\label{pkus}
		\cite{pinkus}
		Let ${\mathcal A}=(A_k)_{k\in{\mathbb N}}\in\calm_d^{\mathbb N}$ be a random, stationary, and ergodic sequence of $d\times d$ invertible matrices with $A_k(i,j) = 0$ for $i > j$ and such that \eqref{f_cond} holds. Then,
		\begin{eqnarray*}
		\gamma_1 = \max_{i\in [d]} {\mathbb E} \left( \log |A_1(i,i)|\right),
		\end{eqnarray*}
		where $[d]:=\{1,\ldots,d\}.$

\begin{proof}
		Define
		\[
		\widehat A_n := \eta_n I_d + A^{-1}U_nV^T .
		\]
		By the commutation assumption, $A$ commutes with $\widehat A_n$ for every $n$, and hence
		\[
		X_n:=A_n\cdots A_1 = A^n \widehat X_n,
		\qquad\text{where } \widehat X_n:=\widehat A_n\cdots \widehat A_1 .
		\]
		Therefore, by submultiplicativity,
		\[
		\|X_n\|\le \|A^n\|\,\|\widehat X_n\|
		\quad\text{and}\quad
		\|\widehat X_n\|=\|A^{-n}X_n\|\le \|A^{-n}\|\,\|X_n\|.
		\]
		Taking $\frac1n\log$ and letting $n\to\infty$, and using Gelfand’s formula
		$\lim_{n\to\infty}\frac1n\log\|A^{\pm n}\|=\log\rho(A^{\pm1})$, gives
		\[
		-\log\rho(A^{-1})+\gamma_1(\widehat\cala)\le \gamma_1(\cala)\le \log\rho(A)+\gamma_1(\widehat\cala).
		\]

		It remains to identify $\gamma_1(\widehat\cala)$.
		Let $K:=\ker(V^T)$, which has dimension $d-m$ (since $\mathrm{rank}(V)=m$). For $x\in K$,
		$V^Tx=0$ and hence $\widehat A_n x=\eta_n x$, so vectors in $K$ grow at rate $\mathbb E\log|\eta_1|$.

		On the quotient space $\mathbb R^d/K$, the induced action is computed via the surjective map
		$q(x)=V^Tx$:
		\[
		q(\widehat A_n x)=V^T(\eta_n x + A^{-1}U_nV^Tx)
		=\bigl(\eta_n I_m + V^TA^{-1}U_n\bigr)\,q(x)
		=\widetilde A_n\,q(x).
		\]
		Thus the cocycle induced on the quotient is precisely $\widetilde\cala$.
		By the block/extension Furstenberg--Kifer lemma (Lemma~\ref{blockt}),
		\[
		\gamma_1(\widehat\cala)=\max\big(\mathbb E\log|\eta_1|,\gamma_1(\widetilde\cala)\big).
		\]
		Substituting into the earlier inequality yields the claimed bounds.
	\end{proof}

	\section{Shape graphs}
	\label{shape}

	We conclude these notes by presenting a generalization of Lemma~\ref{blockt}, partially inspired by the proof of Theorem~A in~\cite{pinkus}. As a prelude to this discussion, we first provide an alternative proof of the upper bound in Lemma~\ref{blockt}, based on the observation that zeros in specific entries of \(A_n\) allow the computation of the Lyapunov exponents in terms of those associated with matrices of simpler structure. We note that the proof previously given for the theorem offers more precise insight into the growth behavior of the blocks and, in particular, can be adapted to establish Proposition~\ref{thm100}.

	For the purpose of this section, let \(\mathcal{A}:=(A_n)_{n\in \mathbb{N}}\) be a random, stationary, and ergodic sequence of matrices as in the statement of Lemma~\ref{blockt}, and write each matrix \(A_n\) as
	\[
	A_n := A_{n,1} + A_{n,2},
	\]
	where \(A_{n,1}\) denotes the block diagonal part and \(A_{n,2}\) the remaining strictly upper triangular component of \(A_n\). \par\noindent{\bf Norm convention.} Throughout Section~\ref{shape} we take $\|\cdot\|$ to be the operator norm induced by the vector $\ell^1$-norm (maximum absolute column sum); we write $\|\cdot\|_1$ when clarity is helpful. By equivalence of norms on the finite-dimensional space $\mathcal M_d$, this choice does not change the Lyapunov exponents and \eqref{f_cond} holds for $\|\cdot\|$ if and only if it holds for $\|\cdot\|_1$. Then
	\[
	X_n = A_{n,1}\cdots A_{1,1}
	+ \sum_{j=1}^n \sum_{1\le i_1 < \cdots < i_j\le n}
	A_{n,1}\cdots A_{i_j+1,1}\, A_{i_j,2}\, A_{i_j-1,1}\cdots A_{i_{j-1}+1,1}\, A_{i_{j-1},2}\,\cdots.
	\]
	It is straightforward to verify that multiplying a block diagonal matrix by a strictly block upper triangular matrix (with the same block sizes) produces a strictly block upper triangular matrix. Moreover, multiplying two strictly upper triangular matrices yields another strictly upper triangular matrix with fewer nonzero superdiagonals—a property reminiscent of the nilpotency of strictly upper triangular matrices. Consequently, all products with \(j > d\) vanish, and therefore
	\[
	X_n = A_{n,1}\cdots A_{1,1}
	+ \sum_{j=1}^d \sum_{1 \leq i_1 < \cdots < i_j \leq n}
	A_{n,1}\cdots A_{i_j+1,1}\, A_{i_j,2}\, A_{i_j-1,1}\cdots A_{i_{j-1}+1,1}\, A_{i_{j-1},2}\,\cdots.
	\]

	Fix $\varepsilon>0$. We will use the following standard Borel--Cantelli estimate (applied to $Y_n=\log^+\|A_{n,2}\|_1$).

	\begin{lemma}\label{lem:bc_log}
		Let $(Y_n)_{n\ge 1}$ be a stationary sequence of nonnegative random variables with $\E(Y_1)<\infty$. Then for every $\varepsilon>0$,
		\[
		\sum_{n\ge 1} \pp(Y_n>\varepsilon n)<\infty,
		\qquad\text{and hence}\qquad
		\pp(Y_n>\varepsilon n\ \text{\rm i.o.})=0.
		\]
	\end{lemma}
	\begin{proof}
		By stationarity, $\pp(Y_n>\varepsilon n)=\pp(Y_1>\varepsilon n)$. Using the tail-integral identity
		$$\E(Y_1)=\int_0^\infty \pp(Y_1>t)\,dt$$ and partitioning the integral into intervals of length $\varepsilon$ gives
		\[
		\sum_{n\ge 1}\pp(Y_1>\varepsilon n)
		\le \frac{1}{\varepsilon}\int_0^\infty \pp(Y_1>t)\,dt
		= \frac{1}{\varepsilon}\E(Y_1)<\infty.
		\]
		The Borel--Cantelli lemma yields the claim.
	\end{proof}

\begin{proposition}
		\label{thm101}
		For given $m\in \nn$ and $(s_1,\ldots,s_m)\in\nn^m,$ let $\cala=(A_n)_{n\in\nn}$ be a tempered sequence of matrices (i.\,e., \eqref{small} holds), each having the shape introduced in \eqref{block_def}. For $i\in [m],$ let $\calb_i:=\big(B_n(i,i)\big)_{n\in \nn},$ and assume that all $B_n(i,i)$ are invertible and all $\calb_i$ are Lyapunov regular (i.\,e., an analogue of \eqref{lr} holds for $\calb_i$). Then, $\gamma_1(\cala)= \max_{j\in [m]} \gamma_1(\calb_j).$
	\end{proposition}

\begin{theorem}
		\label{th-shapes}
		Let $\call$ be a shape set of order $d$ with $k$ elements, and let $(\calv, \calt)_{\call}$ denote its corresponding shape graph. Assume that the shape graph contains only loops of length one (self-loops), and that each nonzero vertex has at most one self-loop. Let $\cala = (A_n)_{n \in \nn}$ be a random, stationary, and ergodic sequence of $d \times d$ matrices, each admitting a decomposition as in~\eqref{A_ni_def} with respect to the shape set $\call$. Assume further that:
		\begin{itemize}
			\item[(i)] For every $1 \leq s \leq k$,
			\[
			\E\big(\log^+ \|A_{1,s}\|\big) < \infty.
			\]
			\item[(ii)] Let $\calh \subset \calv \setminus \{ O_d \}$ be the set of vertices with a self-loop, and let $\call_{\calh}$ be the subset of labels corresponding to their self-loops. For every $s \in \call_\calh$, the matrix $A_{1,s}$ is almost surely invertible, and
			\[
			\E\big(\log^+ \|A_{1,s}^{-1}\|\big) < \infty.
			\]
		\end{itemize}
		Define
		\[
		\beta_s := \lim_{n \to \infty} \frac{1}{n} \log \left\| A_{n,s} A_{n-1,s} \cdots A_{1,s} \right\|, \quad \text{a.s.},
		\]
		and set
		\[
		\beta := \max_{s \in \call_\calh} \beta_s.
		\]

		Then:
		\begin{itemize}
			\item[(a)] The top Lyapunov exponent satisfies
			\[
			\gamma_1(\cala) \leq \beta + \log k.
			\]
			Moreover, if $O_d \in \calv$, define
			\[
			k_* := \max_{v \in \calv \setminus \{O_d\}} \#\bigl\{\, s \in [k] : \calt(v,L_s)\neq O_d \,\bigr\},
			\]
			so that $k_* \le k$ and $k_*$ is the maximal number of labels that keep a nonzero vertex from transitioning to $O_d$ in one step. Then
			\[
			\gamma_1(\cala) \leq \beta + \log k_*.
			\]
			(In particular, if every nonzero vertex has at least one outgoing label that transitions to $O_d$, then $k_* \le k-1$.)
			\item[(b)] Let $\calw \subset \calh\setminus \{O_d\}$ be the set of self-loop vertices $w$ such that every directed path in the shape graph that ends at $w$ avoids loop vertices other than $w$ (equivalently, it does not pass through any vertex in $\calh\setminus\{w\}$). Assume moreover that for each $w\in\calw$, if $s$ denotes the label of its self-loop, then there exists $r=r(w)\ge 1$ such that $\shape{L_s^n}=w$ for all $n\ge r$. If, for every $w \in \calw$ and every $v \in \calv\setminus\{w\}$, we have
			\[
			v \wedge w = O_d,
			\]
			and assume in addition that all matrices $A_{n,s}$ are entrywise nonnegative almost surely (so that monomials with the same shape do not cancel entrywise). Then
			\[
			\max_{s \in \call_\calw} \beta_s \leq \gamma_1(\cala),
			\]
			where $\call_\calw$ denotes the set of labels corresponding to the self-loops of vertices in $\calw$.
		\end{itemize}
	\end{theorem}

\begin{proposition}
		\label{pkus}
		\cite{pinkus}
		Let $\cala=(A_k)_{k\in\nn}\in\calm_d^\nn$ be a random, stationary, and ergodic sequence of $d\times d$ invertible matrices with $A_k(i,j) = 0$ for $i > j$ and such that \eqref{f_cond} holds. Then,
		\beq
		\gamma_1 = \max_{i\in [d]} \E \left( \log |A_1(i,i)|\right),
		\feq
		where $[d]:=\{1,\ldots,d\}.$
	\end{proposition}
	In fact, under the conditions of the proposition, all Lyapunov exponents of $\cala$ can be identified as distinct elements of the multi-set $\{\E \left( \log |A_1(i,i)|\right)\}_{i=1}^d$ (see Corollary~1 in \cite{hen} and item (iv) on p.~130 of \cite[Section~3.2]{arnold}).

	\par
	A conceptual explanation for Proposition~\ref{pkus} is provided by a lemma of Furstenberg and Kifer \cite{fifa} (quoted here as Lemma~\ref{blockt}), which reduces the computation of the top Lyapunov exponent for block-triangular products to that of the diagonal blocks; see, for example, \cite{fifa,kifer,blockas,blockus,tom}.  Extensions appear in several directions, including switched systems \cite{ots, solvable, new} and linear cocycles in bundles \cite[Lemma~3.6]{kifer}; see also \cite[Proposition~1]{blockas}, \cite[Theorem~1.1]{blockus} and \cite[Lemma~4.9]{tom}.  The starting point of this paper is that the same block-triangular viewpoint can be packaged into a practical toolkit that continues to be useful beyond the classical stationary-random setting.  In particular, the resulting bounds are explicit in terms of diagonal-block growth rates and simple combinatorial parameters of the zero pattern, which makes them amenable to computation in matrix-analytic applications.

	\par
	Our contributions are organized around four structured regimes.  Section~\ref{block} begins with deterministic control of upper triangular products: for general tempered sequences (not necessarily random), Proposition~\ref{thm100} gives a two-sided estimate for $\gamma_1$ in terms of diagonal growth rates $\alpha_i^\pm$, and Corollary~\ref{cora} recovers the familiar ``maximum of diagonal exponents'' identity when the relevant diagonal averages converge.  In the same section we develop a deterministic Furstenberg--Kifer-type reduction for block-triangular products under Lyapunov-regularity hypotheses on the diagonal blocks (Proposition~\ref{thm101}), providing a modular step that reduces $\gamma_1$ to diagonal-block data plus an explicit error term.

	Section~\ref{per_sect} illustrates how these reductions turn concrete perturbation models (including rank-one updates) into lower-dimensional or scalar cocycles with computable Lyapunov exponents, and it clarifies when a perturbation changes only the top exponent.  Finally, Section~\ref{shape} introduces shape graphs for sparse decompositions with disjoint supports and limited feedback, proves an energy--entropy bound (Theorem~\ref{th-shapes}) of the form $\gamma_1(\cala)\le \beta+\log k$ (or $\beta+\log k_*$ when $O_d\in\calv$), and provides a short ``how-to'' guide with worked examples in Section~\ref{sec:howto-shapes}, including a transfer-matrix/DAG model that makes the mechanism transparent.

	\section{Block-triangular matrices}
	\label{block}
	We begin with the following inequality concerning the top Lyapunov exponent of a given sequence of upper triangular matrices (cf. \cite[Lemma~3.1.4]{barl}).
	\begin{proposition}
		\label{thm100}
		Let $\cala:=(A_n)_{n\in\nn}$ be a sequence of real, invertible upper triangular $d\times d$ matrices (that is $A_n(j,k)=0$ for $j>k$ and $A_n(j,j)\neq 0,$ $j,k\in [d]$) such that
		\beqn
		\label{small}
		\limsup_{n\to\infty}\frac{1}{n}\log \|A_n\|=0.
		\feqn
		Then,
		\beqn
		\label{D_ineq}
		\max_{j\in [d]} \alpha_j^+\leq \gamma_1
		\leq \max_{j\in [d]} \Big(\alpha_j^+  +\sum_{r=1}^{j-1} \big(\alpha_r^+-\alpha_r^-\big)\Big),
		\feqn
		where
		$
		\alpha_i^-:= \liminf_{n\to\infty} \frac{1}{n} \sum_{k=1}^n \log |A_k(i,i)|$ and $\alpha_i^+:= \limsup_{n\to\infty} \frac{1}{n} \sum_{k=1}^n \log |A_k(i,i)|.
		$
	\end{proposition}

\begin{corollary}
		\label{cora}
		Let the conditions of Proposition~\ref{thm100} hold for $\cala=(A_n)_{n\in\nn}.$ Suppose in addition that $\alpha_i^+=\alpha_i^-$ for all $i\in [d].$
		Then,
		$
		\gamma_1=\max_{j\in [d]} \lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n \log |A_i(j,j)|.
		$
	\end{corollary}

\begin{bmatrix}
				C_n(1,1) & O & \dots & O \\
				O & C_n(2,2) & \dots & O \\
				\vdots & \vdots & \ddots & \vdots \\
				O & O & \dots & C_n(m,m)
			\end{bmatrix}
			,
			\feq
			where $C_n(i,j)$ are $s_i\times s_j$ matrices. Then,
			\beq
			\| X_n \|_F^2 = \| H_n \|_F^2 + \| X_n-H_n \|_F^2 + 2\inner{H_n}{X_n-H_n}_F,
			\feq
		}
		where $\inner{A}{B}_F=\sum_{i,j} A(i,j)B(i,j)$ is the Frobenius inner product of two matrices $A$ and $B,$ provided they have the same shape. Since for any pair $(i,j)\in [m]^2,$ either $H_n(i,j)$ or $(X_n-H_n)(i,j)$ is zero, $\inner{H_n}{X_n-H_n}_F = 0.$
		This implies that $\| X_n \|_F \geq \| H_n \|_F$. Since $H_n$ is block diagonal,
		\[
		\|H_n\|_F^2=\sum_{j=1}^m \|C_n(j,j)\|_F^2.
		\]
		Using the norm equivalence $\|M\|\le \|M\|_F\le \sqrt{d}\,\|M\|$ (valid for all $d\times d$ matrices), each block has the same exponential growth rate whether measured in $\|\cdot\|$ or $\|\cdot\|_F$. Consequently,
		\[
		\lim_{n\to\infty}\frac1n\log\|H_n\|_F=\max_{j\in[m]}\gamma_1(\calb_j)=:\beta,
		\]
		\noindent For each $i\in[m]$, set $\beta_i:=\gamma_1(\calb_i)$, so that $\beta_i\le \beta$.

		and taking $\frac1n\log$ in $\|X_n\|_F\ge \|H_n\|_F$ gives $\beta\le \gamma_1(\cala)$.
		We turn now  to the inverse inequality $\beta \geq \gamma_1(\cala).$ Let $d_0=0$ and $d_j=s_1+\cdots +s_j$ for $j\in[m].$ For $u\in \rr^d,$ let $u_j,$ $j=1,\ldots, m,$ be a vector in $\rr^{s_j}$ with
		\beq
		u_j(i)=u(d_{j-1}+i),\qquad i=1,\ldots, s_j.
		\feq
		Denote by $\|\cdot\|_1$ the $\ell^1$-norm in $\rr^d$ and corresponding matrix norm. Thus, $\|u\|_1=\sum_{i=1}^d |u(i)|$ and $\|A\|_1=\sup_{u\in\rr^d\backslash\{0\}} \frac{\|Au\|_1}{\|u\|_1}=\max_{j\in [d]} \sum_{i=1}^d |A(i,j)|$ for $A \in \calm_d.$ Then, for $u\in \rr^d,$
		\beq
		\|X_n u\|_1&=&\sum_{i=1}^m \Big\|\sum_{j=i}^m C_n (i,j)u_j\Big\|_1\leq \sum_{i=1}^m \sum_{j=i}^m\| C_n (i,j)\|_1\cdot \|u_j\|_1
		\\
		&\leq& \sum_{i=1}^m \sum_{j=i}^m\| C_n (i,j)\|_1\cdot \|u\|_1.
		\feq
		Therefore,
		\beqn
		\label{esta}
		\|X_n \|_1 \leq \sum_{i=1}^m \sum_{j=i}^m\| C_n (i,j)\|_1.
		\feqn
		In view of \eqref{esta}, it suffices to show that for all $i,j\in [m]$ such that $i\leq j,$
		\beqn
		\label{est3}
		\limsup_{n\to\infty} \frac{1}{n}\log \|C_n(i,j)\|_1\leq \beta ,\qquad \as
		\feqn
		To show \eqref{est3}, we adapt an inductive argument employed in the proof of Lemma~3.1.4 in \cite{barr} in order to obtain a version of this estimate for upper-triangular matrices. For $j>i,$ using a convention that $X_0$ is the $d\times d$ identity matrix, we have (cf. (3.10) and (3.11) in \cite{barr}):
		\beqn
		\nonumber
		C_n(i,j)&=&\sum_{t=i+1}^j B_n(i,t)C_{n-1}(t,j)+B_n(i,i)C_{n-1}(i,j)
		\\
		\nonumber
		&=&\sum_{t=i+1}^j B_n(i,t)C_{n-1}(t,j)+B_n(i,i)\sum_{t=i+1}^kB_{n-1}(i,t)C_{n-2}(t,j)
		\\
		\nonumber
		&& \qquad \qquad +B_n(i,i)B_{n-1}(i,i)C_{n-2}(i,j)= \cdots
		\\
		\label{est4}
		&=&
		\sum_{r=0}^{n-1} B_n(i,i)\cdots B_{n-r+1}(i,i)\sum_{t=i+1}^jB_{n-r}(i,t)C_{n-1-r}(t,j).
		\feqn
		Fix some $i^*,j^*\in [m]$ such that $i^*<j^*.$ The inequality in \eqref{est3} is trivially true for $i=j=j^*.$
		Suppose now that \eqref{est3} has been proved for all $(i,j)$ with $j=j^*$ and $i>i^*.$ To finish the proof of the theorem, we will next use a backward induction to show that it is valid for $(i,j)=(i^*,j^*).$ By \eqref{small} and the induction hypothesis, for all $\veps>0$ there exists a constant $D_\veps$ (which depends on $\cala$) such that for all $n\in\nn,$
		\beq
		\|B_n(i,j)\|_1\leq D_\veps e^{n \veps}, \qquad i,j\in [m],
		\feq
		and
		\beq
		\|C_n(i,j^*)\|_1\leq D_\veps e^{n (\beta +\veps)}, \qquad i^*<i\leq j^*.
		\feq
		Furthermore, it follows from \eqref{abs1} that for all $\veps>0$ there exists a constant $G_\veps$ (which depends on $\cala$) such that for any $i\in [m]$ and $k,n\in\nn$ with $k<n,$
		\beqn
		\nonumber
		\big\|B_n(i,i)\cdots B_k(i,i)\big\|_1&=&\big\|B_n(i,i)\cdots B_1(i,i)\cdot B_1^{-1}(i,i)\cdots B_{k-1}^{-1}(i,i)\big\|_1
		\\
		\label{gr}
		&\leq& \big\|B_n(i,i)\cdots B_1(i,i)\big\|_1\cdot \big \|B_1^{-1}(i,i)\cdots B_{k-1}^{-1}(i,i)\big\|_1
		\\
		\nonumber
		&\leq& \big(G_\veps e^{(\beta_i+\veps)n}\big)\cdot\big(G_\veps e^{(-\beta_i+\veps)(k-1)}\big)\leq G_\veps^2 e^{(n-k+1)\beta_i} e^{2n\veps}.
		\feqn
		Therefore, by virtue of \eqref{est4},
		\beq
		\|C_n(i^*,j^*)\|_1&\leq & \sum_{r=0}^{n-1} G_\veps^2 e^{\beta_i r}e^{2n\veps} \sum_{t=i^*+1}^{j^*}D_\veps^2 e^{n \veps} e^{(n-r+1)(\beta+\veps)}\\
		&\leq & nj^* G_\veps^2 D_\veps^2 e^{(4n+1) \veps} e^{(n+1)\beta},
		\feq
		which implies \eqref{est3} with $(i,j)=(i^*,j^*)$ since $\veps>0$ is arbitrary.
	\end{proof}

	\section{Application to matrix perturbations}
	\label{per_sect}

	We next apply the results of Section~\ref{block} to perturbations of linear systems. These perturbations represent an important class of matrix transformations with wide-ranging applications in science and engineering.

	Our first result concerns the product of random rank-one perturbations of a specific form. This proposition can also be viewed somewhat as a generalization of Theorem~5.1 in \cite{rankone}, and contributes to the discussion around Remark~2.4 in the same paper.

	\par

	\begin{proposition} \label{pert_thm}
		Let $(u_n,\eta_n)_{n \in \nn}$ be a random, stationary, and ergodic sequence, where $u_n \in \rr^d$ and $\eta_n \in \rr$. Suppose $v \in \rr^d$ is a fixed vector such that
		\beqn \label{uv_cond}
		\E\bigl(\log |\eta_1|\bigr) \leq \E \bigl(\log |\eta_1 + v^T u_1|\bigr) < \infty.
		\feqn
		\begin{itemize}
			\item[(a)] Let $\cala := (A_n)_{n \in \nn}$ with $A_n := \eta_n I_d + u_n v^T$ satisfying \eqref{f_cond}. Then
			\beqn \label{An_perturbed_exponents}
			\gamma_1(\cala) = \E \bigl(\log |\eta_1 + v^T u_1| \bigr)
			\quad \text{and} \quad
			\gamma_r(\cala) = \E \bigl(\log |\eta_1|\bigr), \quad r > 1.
			\feqn

			\item[(b)] Suppose $A \in \calm_d$ is a non-singular matrix. Let $\cala := (A_n)_{n \in \nn}$ with $A_n := \eta_n A + u_n v^T$ satisfying condition \eqref{f_cond}, and
			\beqn \label{Auv_cond}
			P\bigl(A u_1 v^T = u_1 v^T A\bigr) = 1.
			\feqn
			Then,
			\beq
			- \log \rho(A^{-1}) \leq \gamma_1(\cala) - \E\bigl(\log|\eta_1 + v^T A^{-1} u_1|\bigr) \leq \log \rho(A),
			\feq
			where $\rho(A)$ denotes the spectral radius of the matrix $A$.
		\end{itemize}
	\end{proposition}

	We note that, under mild conditions on the sequence $(\eta_n, u_n)_{n \in \mathbb{N}}$, the inequality~\eqref{uv_cond} can be relaxed, since the result may be applied directly to the inverse cocycle $\cala^{-1}:=(A_n^{-1})_{n\in\nn}$, with part~(a) adjusted accordingly.

	\begin{remark}[Varying right vector: two extendable regimes]
		Consider $A_n=\eta_n I + u_n v_n^{\top}$.

		(i) If $v_n=c_n v$ for a fixed $v\neq 0$, then $A_n=\eta_n I + \tilde u_n v^{\top}$ with
		$\tilde u_n:=c_n u_n$. Hence Proposition~\ref{pert_thm} applies verbatim (under the
		corresponding integrability for $\tilde u_n$), yielding $d-1$ Lyapunov exponents equal to
		$\E\log|\eta_1|$ and the remaining exponent $\E\log|\eta_1+c_1 v^{\top}u_1|$.

		(ii) More generally, if $v_n\in V$ a.s. for a fixed deterministic subspace $V\subset \rr^d$
		of dimension $r$, then $H:=V^{\perp}$ is a common invariant subspace and
		$A_n|_H=\eta_n\,\mathrm{Id}$. Consequently $\E\log|\eta_1|$ is a Lyapunov exponent of
		multiplicity at least $d-r$. The remaining $r$ exponents are those of the induced cocycle
		on the quotient $\rr^d/H$; in general they are not explicit if $v_n$ rotates in $V$.
		Nevertheless, the determinant identity implies
		\[
		\sum_{i=1}^d \gamma_i=(d-1)\E\log|\eta_1|+\E\log|\eta_1+v_1^{\top}u_1|.
		\]
	\end{remark}

	In order to prove Proposition \ref{pert_thm}, we first present a preliminary result that explains how to extract the first few Lyapunov exponents of block upper triangular matrices by Lemma~\ref{blockt}. This extends Corollary~1 in \cite{hen} (cf.\ the paragraph following Proposition~\ref{pkus} in these notes).

	\begin{proposition} \label{block_all_exp}
		Fix $d, m \in \nn$, and let $\caln := (N_n)_{n \in \nn}$ be a random, stationary, and ergodic sequence of matrices
		\beq
		N_n :=
		\begin{bmatrix}
			A_n & * \\
			O & \eta_n I_m
		\end{bmatrix}

Lyapunov exponents play a central role in the analysis of random and chaotic dynamical systems: they quantify sensitivity to initial conditions and, in the linear-cocycle setting, encode long-time growth rates of matrix products; see, e.g., \cite{arnold, barl, boug, fuks, viana}.  Despite their importance, explicit formulas are available only for a comparatively small collection of solvable ensembles---classically, many computations focus on $2\times 2$ products or on highly symmetric distributions (including several Gaussian-type models); see, for instance, \cite{rankone, com, comt, elf, fibbo, keyc, kiev, letac, lima, darm, mann, mark, pinkus} and \cite{adams, ahn, aker, cohen, fort, ens, kargin, newman}.  The purpose of this paper is to extend and reorganize a few ideas from \cite{rankone, fifa, kiev, pinkus} into a small toolkit for obtaining \emph{computable} bounds (and, in favorable cases, explicit values) for Lyapunov exponents of structured matrix products.  From a matrix-analysis viewpoint, the guiding principle is that triangular or block-triangular structure---and more generally prescribed sparsity/zero patterns with limited feedback---localize exponential growth to lower-dimensional diagonal dynamics, while the remaining entries contribute at most an explicitly controlled combinatorial factor.
	Concretely, our bounds are expressed in terms of diagonal-block products together with a directed graph encoding admissible off-diagonal couplings, and are therefore readily computable from a small collection of lower-dimensional norm estimates.
	The framework covers both stationary/ergodic matrix sequences and deterministic sequences satisfying the temperedness condition \eqref{xn}, so it can be read as a matrix-analytic toolkit for growth-rate bounds in structured or sparsely coupled linear systems.

	\par
	For an integer $d\ge 2$, let $\calm_d$ denote the space of $d\times d$ matrices with real-valued entries.  Throughout, we write $\|\cdot\|$ both for a generic norm on ${\mathbb R}^d$ (when the particular choice is immaterial) and for the induced operator norm on $\calm_d$.  Let ${\mathcal A}:=(A_n)_{n\in{\mathbb N}}$ be a sequence in $\calm_d$ such that
	\begin{eqnarray} \label{xn}
	\limsup_{n\to\infty} \frac{1}{n}\log \|X_n\| < \infty\qquad \mbox{\rm where} \qquad
	X_n := A_n \cdots A_1.
	\end{eqnarray}
	We associate to ${\mathcal A}$ the (upper) Lyapunov exponent function $\gamma:{\mathbb R}^d\to{\mathbb R}\cup\{-\infty\}$ defined by
	\begin{eqnarray}
	\label{ld}
	\gamma(v) := \limsup_{n\to\infty}  \frac{1}{n} \log \|X_n v\|.
	\end{eqnarray}
	In particular, we define the \emph{top (upper) Lyapunov exponent} of the sequence ${\mathcal A}$ by
	\begin{eqnarray} \label{gamma1_def}
	\gamma_1({\mathcal A})
	:= \limsup_{n\to\infty}\frac1n\log\|X_n\|
	= \sup_{\|v\|=1}\gamma(v).
	\end{eqnarray}

	This directional growth rate is the basic object we aim to control in the structured regimes considered below.

	\par
	A particularly clean structure emerges when ${\mathcal A}$ is random, stationary, and ergodic and satisfies the standard integrability condition
	\begin{eqnarray} \label{f_cond}
	{\mathbb E}\left( \log^+ \|A_1\| \right) < \infty,
	\end{eqnarray}
	where, for $x>0$, $\log^+ x:=\max(0,\log x)$, with the convention $\log^+0:=0$.  Oseledets' multiplicative ergodic theorem then yields deterministically constant Lyapunov exponents
	\[
	\gamma_1>\cdots>\gamma_\ell,\qquad \ell\le d,
	\]
	together with a corresponding filtration of subspaces
	\[
	{\mathbb R}^d = E_1 \supset E_2 \supset \cdots \supset E_{\ell} \supset E_{\ell+1}:=\{0\},
	\]
	such that the limsup in \eqref{ld} is in fact a limit and $\gamma(v)\in\{\gamma_1,\dots,\gamma_\ell\}$ almost surely, with the value determined by the position of $v$ in the filtration; see \cite[Section~2.1]{barr} and \cite{arnold, barl, viana}.  In particular, the exponents and subspaces are independent of the realization with probability one.

	\par
	Even in this ergodic framework, however, the exponents $\gamma_k$ are rarely available in closed form.  A notable exception occurs for triangular cocycles, where the diagonal entries dictate the asymptotic growth.  In particular, for upper triangular matrices we have:

Lyapunov exponents play a central role in the analysis of random and chaotic dynamical systems: they quantify sensitivity to initial conditions and, in the linear-cocycle setting, encode long-time growth rates of matrix products; see, e.g., \cite{arnold, barl, boug, fuks, viana}. Despite their importance, explicit formulas are available only for a comparatively small collection of solvable ensembles---classically, many computations focus on $2\times 2$ products or on highly symmetric distributions (including several Gaussian-type models); see, for instance, \cite{rankone, com, comt, elf, fibbo, keyc, kiev, letac, lima, darm, mann, mark, pinkus} and \cite{adams, ahn, aker, cohen, fort, ens, kargin, newman}. The purpose of this paper is to extend and reorganize a few ideas from \cite{rankone, fifa, kiev, pinkus} into a small toolkit for obtaining \emph{computable} bounds (and, in favorable cases, explicit values) for Lyapunov exponents of structured matrix products. From a matrix-analysis viewpoint, the guiding principle is that triangular or block-triangular structure---and more generally prescribed sparsity/zero patterns with limited feedback---localize exponential growth to lower-dimensional diagonal dynamics, while the remaining entries contribute at most an explicitly controlled combinatorial factor.
 Concretely, our bounds are expressed in terms of diagonal-block products together with a directed graph encoding admissible off-diagonal couplings, and are therefore readily computable from a small collection of lower-dimensional norm estimates.
 The framework covers both stationary/ergodic matrix sequences and deterministic sequences satisfying the temperedness condition \eqref{xn}, so it can be read as a matrix-analytic toolkit for growth-rate bounds in structured or sparsely coupled linear systems.

\par
 For an integer $d\ge 2$, let $\calm_d$ denote the space of $d\times d$ matrices with real-valued entries. Throughout, we write $\|\cdot\|$ both for a generic norm on ${\mathbb R}^d$ (when the particular choice is immaterial) and for the induced operator norm on $\calm_d$. Let ${\mathcal A}:=(A_n)_{n\in{\mathbb N}}$ be a sequence in $\calm_d$ such that
 \begin{eqnarray} \label{xn}
 \limsup_{n\to\infty} \frac{1}{n}\log \|X_n\| < \infty\qquad \mbox{\rm where} \qquad
 X_n := A_n \cdots A_1.
 \end{eqnarray}
 We associate to ${\mathcal A}$ the (upper) Lyapunov exponent function $\gamma:{\mathbb R}^d\to{\mathbb R}\cup\{-\infty\}$ defined by
 \begin{eqnarray}
 \label{ld}
 \gamma(v) := \limsup_{n\to\infty} \frac{1}{n} \log \|X_n v\|.
 \end{eqnarray}
 In particular, we define the \emph{top (upper) Lyapunov exponent} of the sequence ${\mathcal A}$ by
 \begin{eqnarray} \label{gamma1_def}
 \gamma_1({\mathcal A})
 := \limsup_{n\to\infty}\frac1n\log\|X_n\|
 = \sup_{\|v\|=1}\gamma(v).
 \end{eqnarray}

\par
 A particularly clean structure emerges when ${\mathcal A}$ is random, stationary, and ergodic and satisfies the standard integrability condition
 \begin{eqnarray} \label{f_cond}
 {\mathbb E}\left( \log^+ \|A_1\| \right) < \infty,
 \end{eqnarray}
 where, for $x>0$, $\log^+ x:=\max(0,\log x)$, with the convention $\log^+0:=0$. Oseledets' multiplicative ergodic theorem then yields deterministically constant Lyapunov exponents
 \[
 \gamma_1>\cdots>\gamma_\ell,\qquad \ell\le d,
 \]
 together with a corresponding filtration of subspaces
 \[
 {\mathbb R}^d = E_1 \supset E_2 \supset \cdots \supset E_{\ell} \supset E_{\ell+1}:=\{0\},
 \]
 such that the limsup in \eqref{ld} is in fact a limit and $\gamma(v)\in\{\gamma_1,\dots,\gamma_\ell\}$ almost surely, with the value determined by the position of $v$ in the filtration; see \cite[Section~2.1]{barr} and \cite{arnold, barl, viana}. In particular, the exponents and subspaces are independent of the realization with probability one.

\par
 Even in this ergodic framework, however, the exponents $\gamma_k$ are rarely available in closed form. A notable exception occurs for triangular cocycles, where the diagonal entries dictate the asymptotic growth. In particular, for upper triangular matrices we have:

\label{f_cond}
	\E\left( \log^+ \|A_1\| \right) < \infty,
	\feqn
	where, for $x>0$, $\log^+ x:=\max(0,\log x)$, with the convention $\log^+0:=0$.  Oseledets' multiplicative ergodic theorem then yields

\label{ld}
	\gamma(v) := \limsup_{n\to\infty}  \frac{1}{n} \log \norm{X_n v}.
	\feqn
	In particular, we define the \emph{top (upper) Lyapunov exponent} of the sequence $\cala$ by
	\beqn \label{gamma1_

\label{xn}
	\limsup_{n\to\infty} \frac{1}{n}\log \norm{X_n} < \infty\qquad \mbox{\rm where} \qquad
	X_n := A_n \cdots A_1.
	\feqn
	We associate to $\cala$ the (upper) Lyapunov exponent function $\gamm

and taking $\frac1n\log$ in $\|X_n\|_F\ge \|H_n\|_F$ gives $\beta\le \gamma_1(\cala)$.
 We turn now to the inverse inequality $\beta \geq \gamma_1(\cala).$ Let $d_0=0$ and $d_j=s_1+\cdots +s_j$ for $j\in[m].$ For $u\in \rr^d,$ let $u_j,$ $j=1,\ldots, m,$ be a vector in $\rr^{s_j}$ with
 \beq
 u_j(i)=u(d_{j-1}+i),\qquad i=1,\ldots, s_j.
 \feq
 Denote by $\|\cdot\|_1$ the $\ell^1$-norm in $\rr^d$ and corresponding matrix norm. Thus, $\|u\|_1=\sum_{i=1}^d |u(i)|$ and $\|A\|_1=\sup_{u\in\rr^d\backslash\{0\}} \frac{\|Au\|_1}{\|u\|_1}=\max_{j\in [d]} \sum_{i=1}^d |A(i,j)|$ for $A \in \calm_d.$ Then, for $u\in \rr^d,$
 \beq
 \|X_n u\|_1&=&\sum_{i=1}^m \Big\|\sum_{j=i}^m C_n (i,j)u_j\Big\|_1\leq \sum_{i=1}^m \sum_{j=i}^m\| C_n (i,j)\|_1\cdot \|u_j\|_1
 \\
 &\leq& \sum_{i=1}^m \sum_{j=i}^m\| C_n (i,j)\|_1\cdot \|u\|_1.
 \feq
 Therefore,
 \beqn
 \label{esta}
 \|X_n \|_1 \leq \sum_{i=1}^m \sum_{j=i}^m\| C_n (i,j)\|_1.
 \feqn
 In view of \eqref{esta}, it suffices to show that for all $i,j\in [m]$ such that $i\leq j,$
 \beqn
 \label{est3}
 \limsup_{n\to\infty} \frac{1}{n}\log \|C_n(i,j)\|_1\leq \beta ,\qquad \as
 \feqn
 To show \eqref{est3}, we adapt an inductive argument employed in the proof of Lemma~3.1.4 in \cite{barr} in order to obtain a version of this estimate for upper-triangular matrices. For $j>i,$ using a convention that $X_0$ is the $d\times d$ identity matrix, we have (cf. (3.10) and (3.11) in \cite{barr}):
 \beqn
 \nonumber
 C_n(i,j)&=&\sum_{t=i+1}^j B_n(i,t)C_{n-1}(t,j)+B_n(i,i)C_{n-1}(i,j)
 \\
 \nonumber
 &=&\sum_{t=i+1}^j B_n(i,t)C_{n-1}(t,j)+B_n(i,i)\sum_{t=i+1}^kB_{n-1}(i,t)C_{n-2}(t,j)
 \\
 \nonumber
 && \qquad \qquad +B_n(i,i)B_{n-1}(i,i)C_{n-2}(i,j)= \cdots
 \\
 \label{est4}
 &=&
 \sum_{r=0}^{n-1} B_n(i,i)\cdots B_{n-r+1}(i,i)\sum_{t=i+1}^jB_{n-r}(i,t)C_{n-1-r}(t,j).
 \feqn
 Fix some $i^*,j^*\in [m]$ such that $i^*<j^*.$ The inequality in \eqref{est3} is trivially true for $i=j=j^*.$
 Suppose now that \eqref{est3} has been proved for all $(i,j)$ with $j=j^*$ and $i>i^*.$ To finish the proof of the theorem, we will next use a backward induction to show that it is valid for $(i,j)=(i^*,j^*).$ By \eqref{small} and the induction hypothesis, for all $\veps>0$ there exists a constant $D_\veps$ (which depends on $\cala$) such that for all $n\in\nn,$
 \beq
 \|B_n(i,j)\|_1\leq D_\veps e^{n \veps}, \qquad i,j\in [m],
 \feq
 and
 \beq
 \|C_n(i,j^*)\|_1\leq D_\veps e^{n (\beta +\veps)}, \qquad i^*<i\leq j^*.
 \feq
 Furthermore, it follows from \eqref{abs1} that for all $\veps>0$ there exists a constant $G_\veps$ (which depends on $\cala$) such that for any $i\in [m]$ and $k,n\in\nn$ with $k<n,$
 \beqn
 \nonumber
 \big\|B_n(i,i)\cdots B_k(i,i)\big\|_1&=&\big\|B_n(i,i)\cdots B_1(i,i)\cdot B_1^{-1}(i,i)\cdots B_{k-1}^{-1}(i,i)\big\|_1
 \\
 \label{gr}
 &\leq& \big\|B_n(i,i)\cdots B_1(i,i)\big\|_1\cdot \big \|B_1^{-1}(i,i)\cdots B_{k-1}^{-1}(i,i)\big\|_1
 \\
 \nonumber
 &\leq& \big(G_\veps e^{(\beta_i+\veps)n}\big)\cdot\big(G_\veps e^{(-\beta_i+\veps)(k-1)}\big)\leq G_\veps^2 e^{(n-k+1)\beta_i} e^{2n\veps}.
 \feqn
 Therefore, by virtue of \eqref{est4},
 \beq
 \|C_n(i^*,j^*)\|_1&\leq & \sum_{r=0}^{n-1} G_\veps^2 e^{\beta_i r}e^{2n\veps} \sum_{t=i^*+1}^{j^*}D_\veps^2 e^{n \veps} e^{(n-r+1)(\beta+\veps)}\\
 &\leq & nj^* G_\veps^2 D_\veps^2 e^{(4n+1) \veps} e^{(n+1)\beta},
 \feq
 which implies \eqref{est3} with $(i,j)=(i^*,j^*)$ since $\veps>0$ is arbitrary.
 \end{proof}

\begin{proposition} \label{block_all_exp}
 Fix $d, m \in \nn$, and let $\caln := (N_n)_{n \in \nn}$ be a random, stationary, and ergodic sequence of matrices
 \beq
 N_n :=
 \begin{bmatrix}
 A_n & * \\
 O & \eta_n I_m
 \end{bmatrix},
 \feq
 where $\eta_n \in \rr$ and $A_n \in \calm_d$. Assume that $\caln$ satisfies \eqref{f_cond} (i.e.\ $\E(\log^+\|N_1\|)<\infty$), that $A_1$ is almost surely invertible with $\E(\log^+\|A_1^{-1}\|)<\infty$, and that $\eta_1\neq 0$ almost surely with $\E(\log^+|\eta_1|^{-1})<\infty$. Set $\cala := (A_n)_{n \in \nn}$. Then
 \beq
 \gamma_1(\caln) = \max \bigl( \E(\log | \eta_1 |), \, \gamma_1(\cala) \bigr),
 \feq
 and for any $2 \leq r \leq \min(m,d)$,
 \beq
 && \gamma_r(\caln) = \max_{0\leq \ell \leq r} \left(\sum_{s=1}^\ell \gamma_s(\cala) + (r-\ell) \E(\log | \eta_1|) \right) \\
 && \qquad \qquad - \max_{0\leq \ell \leq r-1} \left(\sum_{s=1}^\ell \gamma_s(\cala) + (r-\ell) \E(\log | \eta_1|) \right).
 \feq
 \end{proposition}
 \begin{proof}
 Recall that a minor of size~$r$ is associated to ordered sequences of indices $J$ and $L$ of length $r$, defined as the determinant of the submatrix formed by selecting the rows indexed by $J$ and the columns indexed by $L$.
 We write $[N]:=\{1,\dots,N\}$ and, for $0\le r\le N$, we let
 $[N]_r:=\{(i_1,\dots,i_r):1\le i_1<\cdots<i_r\le N\}$; we freely identify such increasing $r$--tuples with the underlying $r$--element subsets.
 For $B\subset[m]$ we write $B+d:=\{b+d: b\in B\}\subset\{d+1,\dots,d+m\}$, and for $J_2,L_2\subset\{d+1,\dots,d+m\}$ we interpret
 $I_m[J_2,L_2]:=I_m[J_2-d,\,L_2-d]$.
 For any matrix $A$, let $H_r(A)$ denote the $r$-th compound matrix of $A$, whose entries are all minors of size~$r$. Set $B_n^{(r)} := H_r(N_n)$, and let $\calb_r := (B_n^{(r)})_{n \in \nn}$ be the sequence of $r$-th compound matrices in $\caln$.

\begin{proposition}\label{pert_thm2}
 Fix $m,d\in\mathbb N$, a matrix $V\in\mathcal M_{d\times m}(\mathbb R)$ of rank $m$, and an invertible
 matrix $A\in\mathcal M_{d\times d}(\mathbb R)$.
 Let $(U_n,\eta_n)_{n\ge1}$ be a random stationary ergodic sequence with
 $U_n\in\mathcal M_{d\times m}(\mathbb R)$ and $\eta_n\in\mathbb R$, and define
 \[
 A_n:=\eta_n A + U_nV^T,\qquad
 \widetilde A_n:=\eta_n I_m + V^T A^{-1}U_n .
 \]
 Assume
 \[
 \mathbb P(AU_1V^T = U_1V^T A)=1 .
 \]
 Then, writing $\cala=(A_n)$ and $\widetilde\cala=(\widetilde A_n)$,
 \[
 -\log\rho(A^{-1}) + \max\!\Big(\mathbb E\log|\eta_1|,\,\gamma_1(\widetilde\cala)\Big)
 \;\le\;
 \gamma_1(\cala)
 \;\le\;
 \log\rho(A) + \max\!\Big(\mathbb E\log|\eta_1|,\,\gamma_1(\widetilde\cala)\Big).
 \]
 \end{proposition}

\begin{theorem}
 \label{th-shapes}
 Let $\call$ be a shape set of order $d$ with $k$ elements, and let $(\calv, \calt)_{\call}$ denote its corresponding shape graph. Assume that the shape graph contains only loops of length one (self-loops), and that each nonzero vertex has at most one self-loop. Let $\cala = (A_n)_{n \in \nn}$ be a random, stationary, and ergodic sequence of $d \times d$ matrices, each admitting a decomposition as in~\eqref{A_ni_def} with respect to the shape set $\call$. Assume further that:
 \begin{itemize}
 \item[(i)] For every $1 \leq s \leq k$,
 \[
 \E\big(\log^+ \|A_{1,s}\|\big) < \infty.
 \]
 \item[(ii)] Let $\calh \subset \calv \setminus \{ O_d \}$ be the set of vertices with a self-loop, and let $\call_{\calh}$ be the subset of labels corresponding to their self-loops. For every $s \in \call_\calh$, the matrix $A_{1,s}$ is almost surely invertible, and
 \[
 \E\big(\log^+ \|A_{1,s}^{-1}\|\big) < \infty.
 \]
 \end{itemize}
 Define
 \[
 \beta_s := \lim_{n \to \infty} \frac{1}{n} \log \left\| A_{n,s} A_{n-1,s} \cdots A_{1,s} \right\|, \quad \text{a.s.},
 \]
 and set
 \[
 \beta := \max_{s \in \call_\calh} \beta_s.
 \]

In fact, under the conditions of the proposition, all Lyapunov exponents of ${\mathcal A}$ can be identified as distinct elements of the multi-set $\{{\mathbb E} \left( \log |A_1(i,i)|\right)\}_{i=1}^d$ (see Corollary~1 in \cite{hen} and item (iv) on p.~130 of \cite[Section~3.2]{arnold}).

	\par
	A conceptual explanation for Proposition~\ref{pkus} is provided by a lemma of Furstenberg and Kifer \cite{fifa} (quoted here as Lemma~\ref{blockt}), which reduces the computation of the top Lyapunov exponent for block-triangular products to that of the diagonal blocks; see, for example, \cite{fifa,kifer,blockas,blockus,tom}.  Extensions appear in several directions, including switched systems \cite{ots, solvable, new} and linear cocycles in bundles \cite[Lemma~3.6]{kifer}; see also \cite[Proposition~1]{blockas}, \cite[Theorem~1.1]{blockus} and \cite[Lemma~4.9]{tom}.  The starting point of this paper is that the same block-triangular viewpoint can be packaged into a practical toolkit that continues to be useful beyond the classical stationary-random setting.  In particular, the resulting bounds are explicit in terms of diagonal-block growth rates and simple combinatorial parameters of the zero pattern, which makes them amenable to computation in matrix-analytic applications.

	\par
	Our contributions are organized around four structured regimes.  Section~\ref{block} begins with deterministic control of upper triangular products: for general tempered sequences (not necessarily random), Proposition~\ref{thm100} gives a two-sided estimate for $\gamma_1$ in terms of diagonal growth rates $\alpha_i^\pm$, and Corollary~\ref{cora} recovers the familiar ``maximum of diagonal exponents'' identity when the relevant diagonal averages converge.  In the same section we develop a deterministic Furstenberg--Kifer-type reduction for block-triangular products under Lyapunov-regularity hypotheses on the diagonal blocks (Proposition~\ref{thm101}), providing a modular step that reduces $\gamma_1$ to diagonal-block data plus an explicit error term.

	Section~\ref{per_sect} illustrates how these reductions turn concrete perturbation models (including rank-one updates) into lower-dimensional or scalar cocycles with computable Lyapunov exponents, and it clarifies when a perturbation changes only the top exponent.  Finally, Section~\ref{shape} introduces shape graphs for sparse decompositions with disjoint supports and limited feedback, proves an energy--entropy bound (Theorem~\ref{th-shapes}) of the form $\gamma_1({\mathcal A})\le \beta+\log k$ (or $\beta+\log k_*$ when $O_d\in{\mathcal V}$), and provides a short ``how-to'' guide with worked examples in Section~\ref{sec:howto-shapes}, including a transfer-matrix/DAG model that makes the mechanism transparent.

A conceptual explanation for Proposition~\ref{pkus} is provided by a lemma of Furstenberg and Kifer \cite{fifa} (quoted here as Lemma~\ref{blockt}), which \emph{reduces the computation of the top Lyapunov exponent for block-triangular products to that of the diagonal blocks}. In the present upper-triangular setting, this reduction yields the familiar identification that the top Lyapunov exponent is the maximum among the diagonal-block (here, diagonal-entry) growth rates, i.e. the elements of the multi-set $\{{\mathbb E}(\log|A_1(i,i)|)\}_{i=1}^d$.

A conceptual explanation for Proposition~\ref{pkus} is provided by the following lemma of Furstenberg and Kifer (H. Furstenberg and Y. Kifer, \emph{Random matrix products and measures on projective spaces}, quoted here as Lemma~\ref{blockt}), which \emph{reduces the computation of the top Lyapunov exponent for block-triangular products to that of the diagonal blocks}. We first recall this lemma.

\begin{lemma}
		\cite[Lemma~3.6]{fifa}
		\label{blockt}
		For given $m\in \nn$ and $(s_1,\ldots,s_m)\in\nn^m,$ let $(A_n)_{n\in\nn}$ be a random, stationary ergodic sequence of matrices, each having the shape introduced in \eqref{block_def}. Suppose in addition that for all $i\in [m]$, $B_1(i,i)$ is invertible with probability one, and for all $1\le i\le j\le m$,
		\beq
		\E\big(\log^+ \|B_1(i,j)\|\big)<\infty \qquad \mbox{and} \qquad \E\big(\log^+ \|B_1^{-1}(i,i)\|\big)<\infty.
		\feq
		For $i \in [m],$ let
		\beq
		\beta_i=\lim_{n\to\infty} \frac{1}{n}\log \big\|B_n(i,i)B_{n-1}(i,i)\cdots B_1(i,i)\big\|,\qquad \as
		\feq
		Then, $\gamma_1=\max_{i \in [m]} \beta_i.$
	\end{lemma}

In the present upper-triangular setting, this reduction yields the familiar identification that the top Lyapunov exponent is the maximum among the diagonal-block (here, diagonal-entry) growth rates, i.e. the elements of the multi-set $\{{\mathbb E}(\log|A_1(i,i)|)\}_{i=1}^d$.

\label{pkus}
Pinkus, \emph{(missing title/year in ref\_dict)}
Let ${\mathcal A}=(A_k)_{k\in{\mathbb N}}\in\calm_d^{\mathbb N}$ be a random, stationary, and ergodic sequence of $d\times d$ invertible matrices with $A_k(i,j) = 0$ for $i > j$ and such that
\label{f_cond}
	\E\left( \log^+ \|A_1\| \right) < \infty,
	\feqn
	where, for $x>0$, $\log^+ x:=\max(0,\log x)$, with the convention $\log^+0:=0$.  Oseledets' multiplicative ergodic theorem then yields holds. Then,
\begin{eqnarray*}
\gamma_1 = \max_{i\in [d]} {\mathbb E} \left( \log |A_1(i,i)|\right),
\end{eqnarray*}
where $[d]:=\{1,\ldots,d\}.$,

Question:
Let ${\mathcal A}=(A_k)_{k\in\mathbb N}$ be a random, stationary, and ergodic sequence of invertible real $d\times d$ matrices such that each $A_k$ is upper triangular, i.e. $A_k(i,j)=0$ whenever $i>j$, and assume
\[
\mathbb E\big(\log^+\|A_1\|\big)<\infty,
\]
where $\log^+ x:=\max(0,\log x)$ for $x>0$ and $\log^+0:=0$. If $X_n:=A_n\cdots A_1$, and $\gamma_1$ denotes the top Lyapunov exponent
\[
\gamma_1:=\lim_{n\to\infty}\frac1n\log\|X_n\|
\]
(which exists almost surely and is deterministic under these assumptions), which statement holds for every such sequence?

\[\gamma_1=\max_{i\in[d]}\,\mathbb E\big(\log |A_1(i,i)|\big),\qquad [d]=\{1,\dots,d\}.\]

Choices:
B. \[\gamma_1=\max_{i\in[d]}\,\log\mathbb E\big(|A_1(i,i)|\big),\qquad [d]=\{1,\dots,d\}.\]
C. \[\gamma_1\ge \max_{i\in[d]}\,\mathbb E\big(\log |A_1(i,i)|\big),\qquad [d]=\{1,\dots,d\}.\]
D. \[\gamma_1=\mathbb E\Big(\log \max_{i\in[d]} |A_1(i,i)|\Big),\qquad [d]=\{1,\dots,d\}.\]
E. \[\gamma_1=\max_{i\in[d]}\,\lim_{n\to\infty}\frac1n\log\mathbb E\big|\big(X_n\big)(i,i)\big|,\qquad [d]=\{1,\dots,d\}.\]

equality replaced by one-sided lower bound

max of diagonal Lyapunov rates replaced by expectation of pointwise maximum

almost-sure diagonal growth replaced by growth of expectations with quantifiers moved outside


Question: Let \({\mathcal A}=(A_k)_{k\in\mathbb N}\) be a random, stationary, and ergodic sequence of invertible \(d\times d\) real matrices such that each \(A_k\) is upper triangular, i.e. \(A_k(i,j)=0\) for every \(i>j\), and assume \(\mathbb E(\log^+\|A_1\|)<\infty\), where \(\log^+ x=\max(0,\log x)\). If \(\gamma_1\) denotes the top Lyapunov exponent of the product cocycle \(X_n=A_n\cdots A_1\), namely \(\gamma_1=\limsup_{n\to\infty}\frac1n\log\|X_n\|\), which statement holds?

\(\gamma_1=\max_{i\in[d]}\mathbb E\big(\log |A_1(i,i)|\big)\), where \([d]=\{1,\ldots,d\}\).

Choices:
B. \(\gamma_1=\sum_{i=1}^d \mathbb E\big(\log |A_1(i,i)|\big)\).
C. \(\gamma_1\ge \max_{i\in[d]}\mathbb E\big(\log |A_1(i,i)|\big)\), where \([d]=\{1,\ldots,d\}\).
D. \(\gamma_1=\max_{i\in[d]} \limsup_{n\to\infty}\frac1n\log \big|A_n(i,i)A_{n-1}(i,i)\cdots A_1(i,i)\big|\) for each realization.
E. \(\gamma_1=\max_{i\in[d]}\log \mathbb E\big(|A_1(i,i)|\big)\), where \([d]=\{1,\ldots,d\}\).

max-over-diagonal-block-growth replaced by sum over diagonal entries

equality to the maximal diagonal growth rate weakened to a one-sided lower bound

almost-sure deterministic expectation formula replaced by pathwise realization-level identity

expectation of logarithm replaced by logarithm of expectation


Let $\Lambda_\infty$ be the category whose objects are the natural numbers
$[n]$, $n\ge0$, and whose morphisms are weakly monotone periodic
functions:
\begin{multline*}
  \Lambda_\infty([m],[n]) \\
  = \{ f:\mathbb{Z}\to\mathbb{Z} \mid \text{$f(j)\le f(k)$ if $j\le k$, and
    $f(j+m+1)=f(j)+n+1$} \} .
\end{multline*}
This category was denoted $L$ by Dwyer and Kan \cite{DK}; the notation
$\Lambda_\infty$ is due to Feigin and Tsygan \cite{FT}.

The category $\Lambda_\infty$ has subcategories
\begin{equation*}
  \Lambda_+([m],[n]) = \{ f\in\Lambda_\infty([m],[n]) \mid f(0)\ge0 \} ,
\end{equation*}
denoted $K$ by Dwyer and Kan \cite{DK}, and
\begin{equation*}
  \Delta([m],[n]) = \{ f\in\Lambda_+([m],[n]) \mid f(m)\le n \} .
\end{equation*}

Define the morphisms $\varepsilon^n_i\in\Delta([n{-}1],[n])$,
$0\le i\le n$, $\eta^n_i\in\Delta([n{+}1],[n])$, $0\le i\le n$, and
$\eta^n_{n+1}\in\Lambda_+([n{+}1],[n])$, by
\begin{align*}
  \varepsilon_i^n(j) &=
  \begin{cases}
    j , & 0\le j<i , \\
    j+1 , & i\le j\le n-1 ,
  \end{cases} \intertext{and}
  \eta_i^n(j) &=
  \begin{cases}
    j , & 0\le j\le i , \\
    j-1 , & i<j\le n+1 .
  \end{cases}
\end{align*}
These morphisms satisfy the relations
\begin{align*}
  \varepsilon_i^{n+1} \varepsilon_j^n &= \varepsilon_{j+1}^{n+1} \varepsilon_i^n , \qquad \, \, 0\le i\le j\le n , \\
  \eta_j^n \varepsilon_i^{n+1} &=
                    \begin{cases}
                      \tau_n , & j-i=n+1 , \\
                      \varepsilon_i^n \eta_{j-1}^{n-1} , & 1\le j-i\le n , \\
                      1 , & j-i=0,-1, \\
                      \varepsilon_{i-1}^n \eta_j^{n-1} , & -n-1\le j-i<-1 ,
                    \end{cases} \\
  \eta_j^{n-1} \eta_i^n &= \eta_i^{n-1} \eta_{j+1}^n , \qquad \!\! 0\le i\le j\le n .
\end{align*}
The morphism $\tau_n=\eta^n_{n+1}\varepsilon^{n+1}_0\in\Lambda_\infty([n],[n])$ is the function
\begin{equation*}
  \tau_n(i) = i+1 .
\end{equation*}
We have $\tau_n^{n+1}\varepsilon^n_i=\varepsilon^n_i\tau_{n-1}^n$ and
$\tau_n^{n+1}\eta^n_i=\eta^n_i\tau_{n+1}^{n+2}$. It follows that
$\Lambda_+$ is generated by the morphisms
\begin{equation*}
  \{ \varepsilon_0^n : [n{-}1] \to [n] \}_{n>0} \cup \{ \eta_{n+1}^n : [n{+}1] \to
  [n]\}_{n\ge0} ,
\end{equation*}
and $\Lambda_\infty$ is obtained from $\Lambda_+$ by inverting
$\{ \tau_n^{-1} : [n] \to [n] \}_{n\ge0}$.

There is a contravariant isomorphism of categories $f\mapsto f^\circ$ from
$\Lambda_\infty$ to its opposite $\Lambda^\op_\infty$, which acts on generators by
interchanging $\varepsilon^n_i$ and $\eta^{n-1}_{n-i}$, and fixing
$\tau_n$. In particular, it exchanges the generators $\varepsilon^n_0$ and
$\eta^{n-1}_n$ of $\Lambda_\infty$.

\begin{definition}
  Let $\mathcal{A}$ be a pre-additive category (a category enriched in abelian
  groups).  A \textbf{paracyclic} (respectively \textbf{duplicial} or
  \textbf{simplicial}) module $M_\bullet$ is a presheaf on
  $\Lambda_\infty$ (respectively $\Lambda_+$ or $\Delta$) taking values in $\mathcal{A}$.
\end{definition}

Denote the (co)action of the morphisms $\varepsilon_i^n$ and
$\eta_i^n$ on a duplicial module $M_\bullet$ by
$\partial_{n,i}:M_n\to M_{n-1}$ and $s_{n,i}:M_n\to M_n$, and the (co)action of
the morphisms $\tau_n$ by $t_n:M_n\to M_n$. The relations among the
generators of $\Lambda_+$ become the following relations on duplicial
modules $M_\bullet$:
\begin{align*}
  \partial_{n,k} \partial_{n+1,j} &= \partial_{n,j} \partial_{n+1,k+1} , \qquad \,\, 0\le j\le k\le n , \\
  \partial_{n+1,j} s_{n,k} &=
                    \begin{cases}
                      t_n , & k-j=n+1 , \\
                      s_{n-1,k-1} \partial_{n,j} , & 1\le k-j\le n , \\
                      1 , & k-j=0,-1, \\
                      s_{n-1,k} \partial_{n,j-1} , & -n-1\le k-j<-1 ,
                    \end{cases} \\
  s_{n,j} s_{n-1,k} &= s_{n,k+1} s_{n-1,j} , \qquad \!\! 0\le j\le k\le n .
\end{align*}
In particular,
\begin{align*}
  \partial_{n,i} t_n &=
                \begin{cases}
                  t_{n-1} \partial_{n,i+1} , & 0\le i< n , \\
                  \partial_{n,0} , & i=n ,
                \end{cases} \intertext{and}
  t_{n+1} s_{n,i} &=
                \begin{cases}
                  s_{n,n+1} , & i=0 , \\
                  s_{n,i-1} t_n , & 0< i\le n+1 .
                \end{cases}
\end{align*}
Denote the action of $t_n^{n+1}$ on $M_n$ by $\mathsf{T}$. We see that
$\T_{n-1}\partial_{n,i}=\partial_{n,i}\T_n$ and $\T_{n+1}s_{n,i}=s_{n,i}\T_n$.

An \textbf{idempotent} $(A,p)$ in a pre-additive category $\mathcal{A}$ is a
pair consisting of an object $A$ of $\mathcal{A}$ and $p:A\to A$ an idempotent
endomorphism of $A$, that is, $p^2=p$. If $p:A\to A$ is an idempotent,
then so is $p^\perp=1-p$, and the image of the idempotent $p$ is the
kernel of $p^\perp$ (and vice versa).

Let $M_\bullet$ be a simplicial module. Dold and Puppe \cite{DP}*{Section 3}
define idempotents
\begin{equation*}
  p_n = (1-s_{n-1,0}\partial_{n,1})\ldots(1-s_{n-1,n-1}\partial_{n,n}) : M_n \to M_n .
\end{equation*}
The kernel of the idempotent $p_n:M_n\to M_n$, if it exists, is the
module of degenerate chains $D_n(M)$. The image of $p_n$ is the module
$N_n(M)$ of normalized chains. Provided these modules exist, we have
the direct sum decomposition
\begin{equation*}
  M_n = N_n(M) \oplus D_n(M) .
\end{equation*}

The module $N_n(M)$ is the submodule on which the face maps
$\partial_{n,i}$, $1\le i\le n$, vanish:
\begin{equation*}
  N_n(M) = \bigcap_{i=1}^n \ker\bigl( \partial_{n,i}:M_n\to M_{n-1} \bigr) \subset M_n .
\end{equation*}
Similarly, the module $D_n(M)$ is the sum of the images of the
degeneracy maps $s_{n-1,i}$, $0\le i<n$:
\begin{equation*}
  D_n(M) = \sum_{i=0}^{n-1} \im\bigl( s_{n-1,i}:M_{n-1}\to M_n \bigr) \subset M_n .
\end{equation*}

The following property of a pre-additive category $\mathcal{A}$ was introduced
by Freyd \cite{Freyd} (cf.\ B\"uhler \cite{Buhler}*{Section 7}).
\begin{definition}
  A \textbf{section} of a morphism $f:M\to N$ is a morphism $g:N\to M$
  with
  \begin{equation*}
    fg=1_N .
  \end{equation*}
  A morphism with a section is called a \textbf{retraction}. A
  pre-additive category $\mathcal{A}$ is \textbf{weakly idempotent complete}
  if every retraction has a kernel.
\end{definition}

\begin{lemma}
  If $\mathcal{A}$ is weakly idempotent complete, the idempotent $p_n$ splits,
  implying the existence of the modules $N_n(M)$ and $D_n(M)$.
\end{lemma}
\begin{proof}
  Consider the filtration
  \begin{equation*}
    F_iM_n = \bigcap_{j=n-i+1}^n \ker\bigl( \partial_{n,j}:M_n\to M_{n-1} \bigr) \subset M_n .
  \end{equation*}
  Thus $F_iM_n=M_n$, and $F_nM_n=N_n(M)$.  The existence of $F_iM_n$
  is shown by induction on $i$: $F_{i+1}M_n$ is the kernel of the
  split epimorphism
  \begin{equation*}
    \partial_{n,n-i} : F_iM_n \to F_iM_{n-1} ,
  \end{equation*}
  with section $s_{n-1,n-i-1}:F_iM_{n-1} \to F_iM_n$.
\end{proof}

The following result of Dold and Puppe \cite{DP} is a reformulation of
the theorem of Dold \cite{Dold} and Kan \cite{Kan}. They state the
theorem when $\mathcal{A}$ is abelian, but their proof applies more
generally. (Recall that a pre-additive category is additive if it has
finite sums.)

\begin{definition}
 Let $\mathcal{A}$ be a pre-additive category (a category enriched in abelian
 groups). A \textbf{paracyclic} (respectively \textbf{duplicial} or
 \textbf{simplicial}) module $M_\bullet$ is a presheaf on
 $\Lambda_\infty$ (respectively $\Lambda_+$ or $\Delta$) taking values in $\mathcal{A}$.
\end{definition}

Denote the (co)action of the morphisms $\varepsilon_i^n$ and
$\eta_i^n$ on a duplicial module $M_\bullet$ by
$\partial_{n,i}:M_n\to M_{n-1}$ and $s_{n,i}:M_n\to M_n$, and the (co)action of
the morphisms $\tau_n$ by $t_n:M_n\to M_n$. The relations among the
generators of $\Lambda_+$ become the following relations on duplicial
modules $M_\bullet$:
\begin{align*}
 \partial_{n,k} \partial_{n+1,j} &= \partial_{n,j} \partial_{n+1,k+1} , \qquad \,\, 0\le j\le k\le n , \\
 \partial_{n+1,j} s_{n,k} &=
 \begin{cases}
 t_n , & k-j=n+1 , \\
 s_{n-1,k-1} \partial_{n,j} , & 1\le k-j\le n , \\
 1 , & k-j=0,-1, \\
 s_{n-1,k} \partial_{n,j-1} , & -n-1\le k-j<-1 ,
 \end{cases} \\
 s_{n,j} s_{n-1,k} &= s_{n,k+1} s_{n-1,j} , \qquad \!\! 0\le j\le k\le n .
\end{align*}
In particular,
\begin{align*}
 \partial_{n,i} t_n &=
 \begin{cases}
 t_{n-1} \partial_{n,i+1} , & 0\le i< n , \\
 \partial_{n,0} , & i=n ,
 \end{cases} \intertext{and}
 t_{n+1} s_{n,i} &=
 \begin{cases}
 s_{n,n+1} , & i=0 , \\
 s_{n,i-1} t_n , & 0< i\le n+1 .
 \end{cases}
\end{align*}
Denote the action of $t_n^{n+1}$ on $M_n$ by $\mathsf{T}$. We see that
$\T_{n-1}\partial_{n,i}=\partial_{n,i}\T_n$ and $\T_{n+1}s_{n,i}=s_{n,i}\T_n$.

Let $M_\bullet$ be a simplicial module. Dold and Puppe \cite{DP}*{Section 3}
define idempotents
\begin{equation*}
 p_n = (1-s_{n-1,0}\partial_{n,1})\ldots(1-s_{n-1,n-1}\partial_{n,n}) : M_n \to M_n .
\end{equation*}
The kernel of the idempotent $p_n:M_n\to M_n$, if it exists, is the
module of degenerate chains $D_n(M)$. The image of $p_n$ is the module
$N_n(M)$ of normalized chains. Provided these modules exist, we have
the direct sum decomposition
\begin{equation*}
 M_n = N_n(M) \oplus D_n(M) .
\end{equation*}

The following property of a pre-additive category $\mathcal{A}$ was introduced
by Freyd \cite{Freyd} (cf.\ B\"uhler \cite{Buhler}*{Section 7}).
\begin{definition}
 A \textbf{section} of a morphism $f:M\to N$ is a morphism $g:N\to M$
 with
 \begin{equation*}
 fg=1_N .
 \end{equation*}
 A morphism with a section is called a \textbf{retraction}. A
 pre-additive category $\mathcal{A}$ is \textbf{weakly idempotent complete}
 if every retraction has a kernel.
\end{definition}

\begin{lemma}
 If $\mathcal{A}$ is weakly idempotent complete, the idempotent $p_n$ splits,
 implying the existence of the modules $N_n(M)$ and $D_n(M)$.
\end{lemma}
\begin{proof}
 Consider the filtration
 \begin{equation*}
 F_iM_n = \bigcap_{j=n-i+1}^n \ker\bigl( \partial_{n,j}:M_n\to M_{n-1} \bigr) \subset M_n .
 \end{equation*}
 Thus $F_iM_n=M_n$, and $F_nM_n=N_n(M)$. The existence of $F_iM_n$
 is shown by induction on $i$: $F_{i+1}M_n$ is the kernel of the
 split epimorphism
 \begin{equation*}
 \partial_{n,n-i} : F_iM_n \to F_iM_{n-1} ,
 \end{equation*}
 with section $s_{n-1,n-i-1}:F_iM_{n-1} \to F_iM_n$.
\end{proof}

The following result of Dold and Puppe \cite{DP} is a reformulation of
the theorem of Dold \cite{Dold} and Kan \cite{Kan}. They state the
theorem when $\mathcal{A}$ is abelian, but their proof applies more
generally. (Recall that a pre-additive category is additive if it has
finite sums.)

Let $M_\bullet$ be a simplicial module. Dold and Puppe \cite{DP}*{Section 3}
define idempotents
\begin{equation*}
 p_n = (1-s_{n-1,0}\partial_{n,1})\ldots(1-s_{n-1,n-1}\partial_{n,n}) : M_n \to M_n .
\end{equation*}
The kernel of the idempotent $p_n:M_n\to M_n$, if it exists, is the
module of degenerate chains $D_n(M)$. The image of $p_n$ is the module
$N_n(M)$ of normalized chains. Provided these modules exist, we have
the direct sum decomposition
\begin{equation*}
 M_n = N_n(M) \oplus D_n(M) .
\end{equation*}

\begin{lemma}
 If $\CA$ is weakly idempotent complete, the idempotent $p_n$ splits,
 implying the existence of the modules $N_n(M)$ and $D_n(M)$.
\end{lemma}
\begin{proof}
 Consider the filtration
 \begin{equation*}
 F_iM_n = \bigcap_{j=n-i+1}^n \ker\bigl( \partial_{n,j}:M_n\to M_{n-1} \bigr) \subset M_n .
 \end{equation*}
 Thus $F_iM_n=M_n$, and $F_nM_n=N_n(M)$. The existence of $F_iM_n$
 is shown by induction on $i$: $F_{i+1}M_n$ is the kernel of the
 split epimorphism
 \begin{equation*}
 \partial_{n,n-i} : F_iM_n \to F_iM_{n-1} ,
 \end{equation*}
 with section $s_{n-1,n-i-1}:F_iM_{n-1} \to F_iM_n$.
\end{proof}

We call this the Dold--Kan decomposition.

The purpose of this article is to give a new proof of a result of
Dwyer and Kan \cite{DK}*{Corollary 6.6}.
\begin{theorem}
 \label{DwyerKan}
 Let $\CA$ be a weakly idempotent additive category.
 \begin{enumerate}[i)]
 \item The duplicial module $M_\bullet$ is paracyclic if and only if the
 restriction of the Karoubi operator $\kappa_n$ to $N_n(M)$ is
 invertible for all $n\ge0$.
 \item The duplicial module $M_\bullet$ is cyclic if and only if the
 restriction of the Dwyer--Kan operator $\pi_n$ to $N_n(M)$ is the
 identity for all $n\ge0$.
 \end{enumerate}
\end{theorem}

A simplicial module $M_\bullet$ gives rise to a chain complex: it carries
the differential
\begin{equation*}
 b_n = \sum_{i=0}^n (-1)^i \partial_{n,i} : M_n \to M_{n-1} .
\end{equation*}
We have
\begin{align*}
 b_{n-1}b_n &= \sum_{0\le j\le k\le n-1} (-1)^{j+k} \, \partial_{n-1,k} \partial_{n,j}
 + \sum_{0\le k< j\le n} (-1)^{j+k} \, \partial_{n-1,k} \partial_{n,j} \\
 &= \sum_{0\le j\le k\le n-1} (-1)^{j+k} \, \partial_{n-1,j} \partial_{n,k+1}
 + \sum_{0\le k< j\le n} (-1)^{j+k} \, \partial_{n-1,k} \partial_{n,j} = 0 .
\end{align*}
Note that the restriction of $b_n$ to $N_n(M)$ equals
$\partial_{n,0}$. This gives a formula for the action of the face map
$\delta_n=\partial_{n,0}$ in the Dold--Kan decomposition:
\begin{equation*}
 \delta_n \bigl( s_{n-1,i_1}\dots s_{n-k,i_k} x_{i_1\ldots i_k} \bigr) =
 \begin{cases}
 s_{n-2,i_1-1} \ldots s_{n-k,i_{k-1}-1} x_{i_1\ldots i_k} , & i_k=0 \\
 s_{n-2,i_1-1}\dots s_{n-k-1,i_k-1} ( b_{n-k}x_{i_1\ldots i_k} ) , &
 i_k>0 .
 \end{cases}
\end{equation*}

It was proved by Eilenberg and Mac Lane \cite{EM1} that the complexes
$(M_\bullet,b)$ and $(N_\bullet(M),b)$ are homotopy equivalent. The following
expression for their homotopy is taken from Epstein
\cite{Epstein}*{Proposition 2.3}.
\begin{proposition}
 Let $M_\bullet$ be a simplicial module. The operator
 \begin{equation*}
 \varphi_n = \sum_{i=0}^n (-1)^i s_{n,i} p_{n,i} : M_n \to M_{n+1}
 \end{equation*}
 satisfies $b_{n+1}\varphi_n + \varphi_{n-1}b_n = p_n - 1$.
\end{proposition}
\begin{proof}
 For $0\le i\le n$,
 \begin{align*}
 b_{n+1} s_{n,i} p_{n,i}
 &= \sum_{k=0}^{i-1} (-1)^k \partial_{n+1,k} s_{n,i} p_{n,i}
 + (-1)^i ( \partial_{n+1,i} - \partial_{n+1,i+1} ) s_{n,i} p_{n,i} \\
 & \quad + \sum_{k=i+2}^{n+1} (-1)^k \partial_{n+1,k} s_{n,i} p_{n,i} \\
 &= \sum_{k=0}^{i-1} (-1)^k s_{n-1,i-1} \partial_{n,k} p_{n,i}
 + \sum_{k=i+2}^{n+1} (-1)^k s_{n-1,i} \partial_{n,k-1} p_{n,i} \\
 &= \sum_{k=0}^n (-1)^k s_{n-1,i-1} \partial_{n,k} p_{n,i}
 - (-1)^i s_{n-1,i-1} \partial_{n,i} p_{n,i} \\
 &= s_{n-1,i-1} b_n p_{n,i} + (-1)^i ( p_{n,i-1} - p_{n,i} ) \\
 &= s_{n-1,i-1} p_{n-1,i-1} b_n + (-1)^i ( p_{n,i-1} - p_{n,i} ) .
 \end{align*}
 Summing over $i$, the result follows
\end{proof}

We now come to the main result of this article.
\begin{theorem}
 $\pi_n= p_n \circ \T_n = \T_n \circ p_n$
\end{theorem}
\begin{proof}
 \begin{align*}
 \pi_n &= (-1)^n \partial_{n+1}(\partial_{n+2}s_{n+1}-s_n\partial_{n+1})^ns_n (-1)^{|j|-|i|} \\
 &= t_n^{n+1} + \sum_{k=1}^{n-1} \sum_{0\le i_1<j_1<i_2<j_2< \ldots<i_k<j_k\le n} \\
 & \qquad \bigl( t_n^{i_1} \partial_{n+1} t_{n+1}^{j_1-i_1} s_n \bigr)
 \bigl( t_n^{i_2-j_1-1} \partial_{n+1} t_{n+1}^{j_2-i_2} s_n \bigr) \ldots
 \bigl( t^{i_k-j_{k-1}-1} \partial_{n+1} t_{n+1}^{j_k-i_k} s_n \bigr)
 t_n^{n-j_k} .
 \end{align*}
 If $p<q$, define
 \begin{align*}
 \Pi_{p,q} &= (-1)^{q-p} s_{n-1,p} \partial_{n,q} \\
 &= \bigl( - s_{n-1,p} \partial_{n,p+1} \bigr)
 \bigl( - s_{n-1,p+1} \partial_{n,p+2} \bigr) \ldots
 \bigl( - s_{n-1,q-1} \partial_{n,q} \bigr) .
 \end{align*}
 By definition,
 \begin{equation*}
 \sum_{k=0}^{n-1} \sum_{0\le i_1<j_1<i_2<j_2< \ldots<i_k<j_k\le n} \Pi_{i_1,j_1} \ldots
 \Pi_{i_k,j_k} .
 \end{equation*}
 We have
 \begin{align*}
 (-1)^{j-i} t_n^i \partial_{n+1} t_{n+1}^{j-i} s_n
 &= (-1)^{j-i} t_n^{i-n-1} \partial_{n+1} t_{n+1}^{j-i+n+2} s_n \\
 &= (-1)^{j-i} \partial_{n+1,n-i+1} t_{n+1}^{j+1} s_n
 = (-1)^{j-i} \partial_{n+1,n-i+1} s_{n,n-j} t_n^{j+1} \\
 &= (-1)^{j-i} s_{n-1,n-j} \partial_{n,n-i} t_{n+1}^{j+1} = \Pi_{n-j,n-i} t_n^{j+1} .
 \end{align*}
 It follows that
 \begin{equation*}
 \pi_n = \sum_{k=0}^{n-1} \sum_{0\le i_1<j_1<\ldots\le i_k<j_k\le n}
 \Pi_{n-j_1,n-i_1} \ldots \Pi_{n-j_k,n-i_k} t_n^{n+1} .
 \end{equation*}
 The proof is completed by showing that if $s<p$,
 \begin{equation}
 \label{PiPi}
 \Pi_{p,q} \Pi_{r,s} = \Pi_{r,s} \Pi_{p,q} .
 \end{equation}
 This allows us to reverse the order of the factors in
 $\Pi_{n-j_1,n-i_1} \ldots \Pi_{n-j_k,n-i_k}$:
 \begin{equation*}
 \Pi_{n-j_1,n-i_1} \ldots \Pi_{n-j_k,n-i_k} = \Pi_{n-j_k,n-i_k} \ldots \Pi_{n-j_1,n-i_1} .
 \end{equation*}
 To prove \eqref{PiPi}, we must show that
 \begin{equation*}
 s_{n-1,p} \partial_{n,q} s_{n-1,r} \partial_{n,s}
 = s_{n-1,r} \partial_{n,s} s_{n-1,p} \partial_{n,q} .
 \end{equation*}
 This is a consequence of the simplicial relations:
 \begin{align*}
 s_{n-1,p} \partial_{n,q} s_{n-1,r} \partial_{n,s}
 &= s_{n-1,p} s_{n-2,r} \partial_{n-1,q-1} \partial_{n,s} \\
 &= s_{n-1,r} s_{n-2,p-1} \partial_{n-1,s} \partial_{n,q}
 = s_{n-1,r} \partial_{n,s} s_{n-1,p} \partial_{n,q} .
 \qedhere
 \end{align*}
\end{proof}

Theorem \ref{DwyerKan} is an immediate consequence of this theorem. In
terms of the Dold--Kan decomposition, we see that
\begin{equation*}
 \T_n x = \sum_{k=0}^n \sum_{0\le i_k<\cdots<i_1<n} s^{n-1}_{i_1}\ldots s^{n-k}_{i_k}
 \pi_{n-k}x_{i_1\ldots i_k} .
\end{equation*}
We see that $\T_n$ is invertible on $M_n$ if and only if $\pi_{n-k}$ is
invertible on $N_{n-k}(M)$ for $0\le k\le n$. But $\pi_n$ is invertible on
$N_n(M)$ if and only if $\kappa_n$ is, by the formulas
\begin{equation*}
 \kappa_n^{-1} = \pi_n^{-1} (1+b_{n+1}d_n)^n (1+d_{n-1}b_n)^{n-1}
\end{equation*}
and
\begin{equation*}
 \pi_n^{-1} = \kappa_n^{-n-1} (1+d_{n-1}b_n) .
\end{equation*}

\begin{proof}
  The proof is by induction on $n$: the case $n=0$ is trivial, since
  $M_0=N_0(M)$. The induction step is obtained by writing $x\in M_n$ as
  \begin{equation*}
    p_nx + (1-p_n)x = p_nx + \sum_{i=1}^n s_{n-1,i-1}\partial_{n,i}p_{n,i}x .
  \end{equation*}
  Since $\partial_{n,i}p_{n,i}x\in M_{n-1}$, the result follows.
\end{proof}

We call this the Dold--Kan decomposition.

\begin{definition}
  Let $M_\bullet$ be a duplicial module. The \textbf{Karoubi operator} is
  the endomorphism of $M_n$
  \begin{equation*}
    \kappa_n = (-1)^n \bigl( \partial_{n+1,0} s_{n,n+1} - s_{n-1,n}\partial_{n,0} \bigr) .
  \end{equation*}
The \textbf{Dwyer--Kan operator} is the endomorphism of $M_n$
  \begin{equation*}
    \pi_n = (-1)^n \partial_{n+1,0} \kappa_{n+1}^n s_{n,n+1} .
  \end{equation*}
\end{definition}

The definitions of these operators are simplified when they are
expressed in terms of the operators $\delta_n=\partial_{n,0}$ and
$\sigma_n=(-1)^ns_{n,n+1}$:
\begin{equation*}
  \kappa_n = \delta_{n+1} \sigma_n + \sigma_{n-1}\delta_n
\end{equation*}
and
\begin{equation*}
  \pi_n = \delta_{n+1} \kappa_{n+1}^n \sigma_n .
\end{equation*}

The restriction of $\pi_n$ to $N_n(M)$ was studied by Dwyer and Kan
\cite{DK}, and further studied in the special case of cyclic modules
by Cuntz and Quillen \cite{CQ}.

The purpose of this article is to give a new proof of a result of
Dwyer and Kan \cite{DK}*{Corollary 6.6}.
\begin{theorem}
  \label{DwyerKan}
  Let $\mathcal{A}$ be a weakly idempotent additive category.
  \begin{enumerate}[i)]
  \item The duplicial module $M_\bullet$ is paracyclic if and only if the
    restriction of the Karoubi operator $\kappa_n$ to $N_n(M)$ is
    invertible for all $n\ge0$.
  \item The duplicial module $M_\bullet$ is cyclic if and only if the
    restriction of the Dwyer--Kan operator $\pi_n$ to $N_n(M)$ is the
    identity for all $n\ge0$.
  \end{enumerate}
\end{theorem}

Unlike in the earlier work of Dwyer and Kan \cite{DK} and Cuntz and
Quillen \cite{CQ}, we consider the action of the operators $\kappa_n$ and
$\pi_n$ on all of $M_n$, and not only on $N_n(M)$. We show by direct
calculation that the operator $\pi_n$ equals the composition of the
projection $p_n$ of Dold and Puppe with the operator $\T_n$. In
particular, for cyclic modules, $\pi_n=p_n$.

Dwyer and Kan cast the Dold--Kan theorem as an equivalence between the
category of duplicial modules and the category of duchain complexes.
\begin{definition}
  A \textbf{duchain complex} is a graded module $M_\bullet$ with a pair of
  differentials $b_n:M_n\to M_{n-1}$ and $d_n:M_n\to M_{n+1}$.
\end{definition}

A chain complex $(V_\bullet,b)$ is a degenerate duchain complex, whose extra
differential $d$ vanishes. Thus, there is a natural embedding of the
category of simplicial modules to the category of duplicial (in fact,
cyclic) modules, given by taking the duplicial structure on the
simplicial module $M_\bullet$ induced by this degenerate duchain structure
on $N_\bullet(M)$. In this way, we obtain a new formula for the idempotent
$p_n$ of Dold and Puppe on $M_n$: it equals the Dwyer--Kan operator
$\pi_n$.

The proof of Theorem~\ref{DoldKan} is "by induction on $n$": for $n=0$ it is trivial since "$M_0=N_0(M)$." For the induction step, one writes any $x\in M_n$ as
\[
  p_nx + (1-p_n)x = p_nx + \sum_{i=1}^n s_{n-1,i-1}\partial_{n,i}p_{n,i}x .
\]
Then one notes that "Since $\partial_{n,i}p_{n,i}x\in M_{n-1}$, the result follows."

To prove the main theorem, we argue by induction on $n$: for $n=0$ it is trivial since "$M_0=N_0(M)$." For the induction step, one writes any $x\in M_n$ as
\[
  p_nx + (1-p_n)x = p_nx + \sum_{i=1}^n s_{n-1,i-1}\partial_{n,i}p_{n,i}x .
\]
Then one notes that "Since $\partial_{n,i}p_{n,i}x\in M_{n-1}$, the result follows."

\label{DoldKan}
  Let $\mathcal{A}$ be a weakly idempotent additive category. Let $M_\bullet$ be a
  simplicial module. Every element of $M_n$ has a unique decomposition
  \begin{equation*}
    x = \sum_{k=0}^n \sum_{0\le i_k<\cdots<i_1<n} s_{n-1,i_1}\ldots s_{n-k,i_k}x_{i_1\ldots i_k} ,
  \end{equation*}
  where $x_{i_1\ldots i_k} \in N_{n-k}(M)$.,

Question:
Let $\mathcal{A}$ be a weakly idempotent additive category, and let $M_\bullet$ be a simplicial module in $\mathcal{A}$ with degeneracy maps $s_{m,i}:M_m\to M_{m+1}$. For each $m$, let
\[
p_m=(1-s_{m-1,0}\partial_{m,1})\cdots(1-s_{m-1,m-1}\partial_{m,m}):M_m\to M_m,
\]
and let $N_m(M)$ denote the image of $p_m$ (the normalized chains). Which statement holds for each $n\ge 0$ and each element $x\in M_n$?

Every element $x\in M_n$ admits a unique decomposition
\[
x=\sum_{k=0}^n\ \sum_{0\le i_k<\cdots<i_1<n} s_{n-1,i_1}\cdots s_{n-k,i_k}\,x_{i_1\ldots i_k},
\]
where for each multi-index $(i_1,\ldots,i_k)$ one has $x_{i_1\ldots i_k}\in N_{n-k}(M)$.

Choices:
B.
Every element $x\in M_n$ admits a decomposition
\[
x=\sum_{k=0}^n\ \sum_{0\le i_k\le\cdots\le i_1<n} s_{n-1,i_1}\cdots s_{n-k,i_k}\,x_{i_1\ldots i_k},
\]
where for each multi-index $(i_1,\ldots,i_k)$ one has $x_{i_1\ldots i_k}\in N_{n-k}(M)$, and this decomposition is unique.
C.
Every element $x\in M_n$ can be written in the form
\[
x=\sum_{k=0}^n\ \sum_{0\le i_k<\cdots<i_1<n} s_{n-1,i_1}\cdots s_{n-k,i_k}\,x_{i_1\ldots i_k},
\]
where for each multi-index $(i_1,\ldots,i_k)$ one has $x_{i_1\ldots i_k}\in N_{n-k}(M)$.
D.
For each $n\ge 0$ there exists a choice of normalized coefficients $x_{i_1\ldots i_k}\in N_{n-k}(M)$, depending only on $n$ and not on the particular element $x\in M_n$, such that every element $x\in M_n$ admits a unique decomposition
\[
x=\sum_{k=0}^n\ \sum_{0\le i_k<\cdots<i_1<n} s_{n-1,i_1}\cdots s_{n-k,i_k}\,x_{i_1\ldots i_k}.
\]
E.
Every element $x\in M_n$ admits a unique decomposition
\[
x=\sum_{k=0}^n\ \sum_{0\le i_k<\cdots<i_1<n} s_{n-1,i_1}\cdots s_{n-k,i_k}\,x_{i_1\ldots i_k},
\]
where for each multi-index $(i_1,\ldots,i_k)$ one has $x_{i_1\ldots i_k}\in M_{n-k}$.

strictly increasing multi-index condition

dependence of coefficients on the chosen element x

normalized-chain condition on coefficients


\begin{theorem}\label{Th13}
Let $2\le p <\infty,\;0<s<1$ and $0\le \Lambda \le 1$. Suppose $\Omega \in \mathbb{H}^n$ is a bounded open set and $u \in HW_{{\rm{loc}}}^{1,p}\left( \Omega \right) \cap{ L_{sp}^{p - 1}\left( {{\mathbb{H}^n}} \right)} $ is a weak solution of \eqref{eq0}. Then $u \in C_{loc}^\gamma \left( \Omega  \right)$ for any $0 < \gamma  < 1$. Moreover, for any $0 < \gamma  < 1$ and any Kor\'{a}nyi ball ${B_{2R}}\left( {{\xi _0}} \right) \subset  \subset \Omega $ with $0<R<1$, there exists a positive constant $c=c(Q,p,s,\gamma)$ such that
\begin{equation}\label{eq482}
  \mathop {\sup }\limits_{\xi  \ne \eta  \in {B_{\frac{R}{2}}}\left( {{\xi _0}} \right)} \frac{{\left| {u\left( \xi  \right) - u\left( \eta  \right)} \right|}}{{{{\left\| {{\eta ^{ - 1}} \circ \xi } \right\|}^\gamma _{\mathbb{H}^n}}}} \le \frac{c}{{{R^\gamma }}}\left( {{{{\left\| u \right\|}_{{L^\infty }\left( {{B_R}\left( {{\xi _0}} \right)} \right)}}} + {\rm{Tai}}{{\rm{l}}_{p - 1,sp,p}}\left( {u;{\xi _0},R} \right)} \right),
\end{equation}
where the definition of ${\rm{Tai}}{{\rm{l}}_{p - 1,sp,p}}\left( {u;{\xi _0},R} \right)$ see \eqref{eq26} below.
\end{theorem}

\label{sec-1}

The study of regularity theory for degenerate elliptic equations has long stood as a cornerstone of modern analysis, tracing its intellectual lineage to the groundbreaking contributions of De Giorgi, Nash, and Moser. In the context of the Heisenberg group $\mathbb{H}^n$, a paradigmatic example of a sub-Riemannian manifold, the analysis of quasilinear degenerate elliptic equations of $p$-sub-Laplacian type has undergone profound development. In the pioneering work \cite{CL97}, Capogna established $C^{1,\alpha}$-regularity for the case $p=2$. Subsequently, Mukherjee and Zhong\cite{MZ21} extended this result to quasilinear degenerate elliptic equations with $p$-growth conditions for all $1<p<\infty$ via De Giorgi's method. Further progress includes: fractional differentiability estimates for nonlinear non-differentiable degenerate elliptic equations obtained by Zhang and Niu\cite{ZN22}; ${C^{\alpha }}$-regularity of minimizers for generalized Orlicz functionals encompassing $p,q$-growth conditions established by Zhang and Niu\cite{ZN20}; weak differentiability of weak solutions under the constraint $1<p<4$ for quasilinear equations with $p,q$-growth conditions proved by Zhang and Li\cite{ZL23}. A sufficient condition excluding the Lavrentiev phenomenon for non-autonomous integral functionals with $p,q$-growth condition in the same setting provided by Zhang and Niu\cite{ZN231}. For additional developments and comprehensive references, we refer the reader to \cite{ZN22} and the works cited therein.

The regularity theory for purely nonlocal equations involving fractional powers of the $p$-Laplacian on the Heisenberg group has witnessed remarkable progress in recent years, spurred by pivotal advancements such as the introduction of nonlocal tailing mechanisms and the establishment of nonlocal Harnack and H\"{o}lder regularity estimates. These contributions, as reported in Roncal and Thangavelu\cite{RT16}, Frank, Lieb, and Seiringer\cite{FLS08}, Ferrari and Franchi\cite{FF15}, Ferrari, Miranda, Pallara, Pinamonti and Sire\cite{FMPPS18}, Frank, Gonzalez, Monticelli and Tan\cite{FGMT15}, Manfredini, Palatucci, Piccinini and Polidoro\cite{MPPP23}, and Palatucci and Piccinini\cite{PP22}, collectively constitute a transformative body of work reshaping the analytical foundations of the field. Moreover, Palatucci and Piccinini in 2022 (see \cite{PP22}) and 2023 (see \cite{PP23}) proposed an open problem how to determine the regularity of nonlocal double-phase equations in the Heisenberg group. Fang, Zhang and Zhang provided a solution in \cite{FZZ24}.

However, the analysis of equations combining local and nonlocal operators-referred to as "mixed" equations-presents a formidable challenge due to the intricate interplay between classical diffusion processes and long-range interactions. While the Euclidean setting has witnessed rapid progress, with results ranging from local boundedness (see Garain and Kinnunen\cite{GK22}) to higher-order H\"{o}lder estimates (see Garain and Lindgren\cite{GL23}), extending such theories to the non-commutative geometry of the Heisenberg group demands overcoming profound technical obstacles, including the absence of embedding theorems and the inherent anisotropy of the Carnot-Carath\'{e}odory metric.

Recently, Zhang, Niu and Wu in \cite{ZNW25} proved that both the derivative in the vertical direction and the derivative in the horizontal direction of weak solutions $u \in H{W^{1,p}}\left( \Omega  \right)$ to mixed sub-Laplace and fractional sub-Laplace equations and $f \in {L^2}\left( \Omega  \right)$ belonging to the first-order local Sobolev space $HW_{loc}^{1,p}\left( \Omega  \right)$. Furthermore, under the assumption $f \in H{W^{m,p}}\left( \Omega  \right)$, it follows that $u \in HW_{loc}^{m + 2,p}\left( \Omega  \right)$. Subsequently, Zhang and Niu\cite{ZN26} achieved a significant breakthrough by establishing fundamental regularity properties for weak solutions to mixed local and nonlocal degenerate elliptic equations in $\mathbb{H}^n$. Specifically, they demonstrated the local boundedness of weak subsolutions, the H\"{o}lder continuity of weak solutions, and critically, both the Harnack inequality and the weak Harnack inequality. These results affirm that the classical difference and De Giorgi-Nash-Moser theory can be fruitfully extended to this hybrid setting, laying a robust foundation for the qualitative analysis of such equations. While H\"{o}lder continuity of the solution $u$ has been established, a central open question remains: does the horizontal gradient ${{\nabla _H}u}$ exhibit H\"{o}lder continuity, i.e., do solutions belong to the $C^{1,\alpha}$ class? This problem is exceptionally intricate in the context of degenerate ($p>2$) equations coupled with mixed operators, as standard linearization techniques prove inadequate, and the interplay between the local $p$-Laplacian structure and the nonlocal fractional term obfuscates the differentiation of the equation.

In this paper, we bridge this gap by investigating the $C^{1,\alpha}$-regularity of weak solutions to mixed local and nonlocal degenerate elliptic equations in the Heisenberg group. Building on the Harnack estimates and tail control developed in \cite{ZN26}, we aim to prove that under suitably structured assumptions, weak solutions not only exhibit H\"{o}lder continuity but also possess higher smoothness properties with their horizontal derivatives satisfying a H\"{o}lder condition. This result will extend the classical $C^{1,\alpha}$ estimates for $p$-sub-Laplace equations in $\mathbb{H}^n$ to the more intricate landscape of mixed local-nonlocal diffusion.

In this paper we focus on the analysis of the mixed local and nonlocal degenerate elliptic equations
\begin{equation}\label{eq0}
 - \Delta _{{{\mathbb{H}},p}}u + \Lambda{\left( { - \Delta _{{{\mathbb{H}},p}}} \right)^s}u = 0\;{\rm{in}}\;\Omega ,
\end{equation}
where $2\le p< \infty,\;0<s<1,\;0\le\Lambda \le 1 $, and $\Omega$ is a bounded open subset in the Heisenberg group  ${{\mathbb{H}}^n},\;n\ge1$. Here the $p$-sub-Laplace operator on the Heisenberg group is defined by
\begin{equation}\label{eq12}
 - {\Delta _{{\mathbb{H}},p}}u = di{v_H}\left( {{{\left| {{\nabla _H}u} \right|}^{p - 2}}{\nabla _H}u} \right),
\end{equation}
while the fractional $p$-sub-Laplace operator on the Heisenberg group is given by
\begin{equation}\label{eq13}
{\left( { - {\Delta _{{\mathbb{H}},p}}} \right)^s}u = {\rm{P}}{\rm{.V}}{\rm{.}}\int_{{{\mathbb{H}}^n}} {\frac{{{{\left| {u\left( \xi  \right) - u\left( \eta  \right)} \right|}^{p - 2}}\left( {u\left( \xi  \right) - u\left( \eta  \right)} \right)}}{{{{\left\| {{\eta ^{ - 1}} \circ \xi } \right\|}^{Q + sp}_{\mathbb{H}^n}}}}d\eta },
\end{equation}
where P.V. signifies the Cauchy principal value and $Q=2n+2$ is the homogeneous dimension of ${{\mathbb{H}}^n}$.

The main results are as follows:

The regularity theory for purely nonlocal equations involving fractional powers of the $p$-Laplacian on the Heisenberg group has witnessed remarkable progress in recent years, spurred by pivotal advancements such as the introduction of nonlocal tailing mechanisms and the establishment of nonlocal Harnack and H\"{o}lder regularity estimates. These contributions, as reported in Roncal and Thangavelu\cite{RT16}, Frank, Lieb, and Seiringer\cite{FLS08}, Ferrari and Franchi\cite{FF15}, Ferrari, Miranda, Pallara, Pinamonti and Sire\cite{FMPPS18}, Frank, Gonzalez, Monticelli and Tan\cite{FGMT15}, Manfredini, Palatucci, Piccinini and Polidoro\cite{MPPP23}, and Palatucci and Piccinini\cite{PP22}, collectively constitute a transformative body of work reshaping the analytical foundations of the field. Moreover, Palatucci and Piccinini in 2022 (see \cite{PP22}) and 2023 (see \cite{PP23}) proposed an open problem how to determine the regularity of nonlocal double-phase equations in the Heisenberg group. Fang, Zhang and Zhang provided a solution in \cite{FZZ24}.

Recently, Zhang, Niu and Wu in \cite{ZNW25} proved that both the derivative in the vertical direction and the derivative in the horizontal direction of weak solutions $u \in H{W^{1,p}}\left( \Omega \right)$ to mixed sub-Laplace and fractional sub-Laplace equations and $f \in {L^2}\left( \Omega \right)$ belonging to the first-order local Sobolev space $HW_{loc}^{1,p}\left( \Omega \right)$. Furthermore, under the assumption $f \in H{W^{m,p}}\left( \Omega \right)$, it follows that $u \in HW_{loc}^{m + 2,p}\left( \Omega \right)$. Subsequently, Zhang and Niu\cite{ZN26} achieved a significant breakthrough by establishing fundamental regularity properties for weak solutions to mixed local and nonlocal degenerate elliptic equations in $\mathbb{H}^n$. Specifically, they demonstrated the local boundedness of weak subsolutions, the H\"{o}lder continuity of weak solutions, and critically, both the Harnack inequality and the weak Harnack inequality. These results affirm that the classical difference and De Giorgi-Nash-Moser theory can be fruitfully extended to this hybrid setting, laying a robust foundation for the qualitative analysis of such equations. While H\"{o}lder continuity of the solution $u$ has been established, a central open question remains: does the horizontal gradient ${{\nabla _H}u}$ exhibit H\"{o}lder continuity, i.e., do solutions belong to the $C^{1,\alpha}$ class? This problem is exceptionally intricate in the context of degenerate ($p>2$) equations coupled with mixed operators, as standard linearization techniques prove inadequate, and the interplay between the local $p$-Laplacian structure and the nonlocal fractional term obfuscates the differentiation of the equation.

In this paper, we bridge this gap by investigating the $C^{1,\alpha}$-regularity of weak solutions to mixed local and nonlocal degenerate elliptic equations in the Heisenberg group. Building on the Harnack estimates and tail control developed in \cite{ZN26}, we aim to prove that under suitably structured assumptions, weak solutions not only exhibit H\"{o}lder continuity but also possess higher smoothness properties with their horizontal derivatives satisfying a H\"{o}lder condition. This result will extend the classical $C^{1,\alpha}$ estimates for $p$-sub-Laplace equations in $\mathbb{H}^n$ to the more intricate landscape of mixed local-nonlocal diffusion.

In this paper we focus on the analysis of the mixed local and nonlocal degenerate elliptic equations
\begin{equation}\label{eq0}
 - \Delta _{{{\mathbb{H}},p}}u + \Lambda{\left( { - \Delta _{{{\mathbb{H}},p}}} \right)^s}u = 0\;{\rm{in}}\;\Omega ,
\end{equation}
where $2\le p< \infty,\;0<s<1,\;0\le\Lambda \le 1 $, and $\Omega$ is a bounded open subset in the Heisenberg group ${{\mathbb{H}}^n},\;n\ge1$. Here the $p$-sub-Laplace operator on the Heisenberg group is defined by
\begin{equation}\label{eq12}
 - {\Delta _{{\mathbb{H}},p}}u = di{v_H}\left( {{{\left| {{\nabla _H}u} \right|}^{p - 2}}{\nabla _H}u} \right),
\end{equation}
while the fractional $p$-sub-Laplace operator on the Heisenberg group is given by
\begin{equation}\label{eq13}
{\left( { - {\Delta _{{\mathbb{H}},p}}} \right)^s}u = {\rm{P}}{\rm{.V}}{\rm{.}}\int_{{{\mathbb{H}}^n}} {\frac{{{{\left| {u\left( \xi \right) - u\left( \eta \right)} \right|}^{p - 2}}\left( {u\left( \xi \right) - u\left( \eta \right)} \right)}}{{{{\left\| {{\eta ^{ - 1}} \circ \xi } \right\|}^{Q + sp}_{\mathbb{H}^n}}}}d\eta },
\end{equation}
where P.V. signifies the Cauchy principal value and $Q=2n+2$ is the homogeneous dimension of ${{\mathbb{H}}^n}$.

The main results are as follows:

\begin{remark}
In contrast to \cite{ZN26}, where the H\"older index $\gamma \in (0,\frac{sp}{p-1})$ relies on the parameters $s$ and $p$, Theorem \ref{Th13} establishes a parameter-independent H\"older regularity with $\gamma \in (0,1)$. This holds under more general nonlinear frameworks and is particularly advantageous for subsequent regularity analysis.
\end{remark}

\begin{proposition}\label{Pro41}
Let $2\le p <\infty,\;0<s<1$ and $0\le \Lambda \le 1$. Assume $u \in HW_{{\rm{loc}}}^{1,p}\left( {{B_2}\left( {{\xi _0}} \right) \cap L_{sp}^{p - 1}\left( {{\mathbb{H}^n}} \right)} \right)$ is a weak solution of \eqref{eq0} in ${{B_2}\left( {{\xi _0}} \right)}$ satisfying
\begin{equation}\label{eq41}
{\left\| u \right\|_{{L^\infty }\left( {{B_1}\left( {{\xi _0}} \right)} \right)}} \le 1,\int_{{\mathbb{H}^n}\backslash {B_1}\left( {{\xi _0}} \right)} {\frac{{{{\left| {u\left( \eta \right)} \right|}^{p - 1}}}}{{\left\| \eta \right\|_{{\mathbb{H}^n}}^{Q + sp}}}d\eta } \le 1.
\end{equation}
If $0<h_0<\frac{1}{10}$ and $R$ satisfy $4{h_0}<R\le 1-5{h_0}$, and ${\nabla _H}u \in {L^q}\left( {{B_{R + 4{h_0}}}\left( {{\xi _0}} \right)} \right)$ for some $q \ge p$, then
\begin{equation}\label{eq42}
 \mathop {\sup }\limits_{0 < \left| h \right| < {h_0}} \int_{{B_{R - {4h_0}}}} {{{\left| {\frac{{\Delta _{Z,h}^2u}}{{{{\left| h \right|}^{\frac{1}{{q + 1}} + 1}}}}} \right|}^{q + 1}}d\xi } \le c(1+\Lambda) \left( {\int_{{B_{R + {4h_0}}}} {{{\left| {{\nabla _H}u} \right|}^q}d\xi } + 1} \right),
\end{equation}
where $c=c(Q,h_0,p,q,s)>0$.
\end{proposition}

For simplicity, we denote $u_R$ by $u$. Fix $0<\gamma<1$ and choose $i_\infty \in \mathbb{N}\setminus {0}$ such that
\begin{equation}\label{eq492}
1 - \gamma > \frac{Q}{{p + {i_\infty }}}.
\end{equation}
For $i = 0, \cdots ,{i_\infty }$, we define
\[{q_i} = p + i,\]
and
\[{h_0} = \frac{1}{{112{i_\infty }}},\;{R_i} = \frac{7}{8} - 4{h_0} - 14{h_0}i,\]
so
\[{R_0} + 4{h_0} = \frac{7}{8},\;{R_{{i_\infty }}} + 4{h_0} = \frac{3}{4},\;4{h_0} <R_i<1-5{h_0} .\]
By using Proposition \ref{Pro41} with
\[R = {R_i},q = {q_i},\;i = 0, \cdots ,{i_\infty },\]
and we have by \eqref{eq485}
\begin{equation}\label{eq486}
\mathop {\sup }\limits_{0 < \left| h \right| < {h_0}} \int_{{B_{R_0 - {4h_0}}}} {{{\left| {\frac{{\Delta _{Z,h}^2u}}{{{{\left| h \right|}^{\frac{1}{{{q_1}}} + 1}}}}} \right|}^{{q_1}}}d\xi } \le c(1 + \Lambda )\left( {\int_{{B_{\frac{7}{8}}}} {{{\left| {{\nabla _H}u} \right|}^p}d\xi } + 1} \right) \le C\left( {Q,p,s,\gamma} \right).
\end{equation}
Consider ${R_i} - 10{h_0} = {R_{i + 1}} + 4{h_0}$ for $i = 0, \cdots ,{i_\infty }$, we use Lemma \ref{Le213} in \eqref{eq486} to get
\begin{equation}\label{eq487}
\int_{{B_{{R_1} + 4{h_0}}}} {{{\left| {{\nabla _H}{u}} \right|}^{{q_1}}}d\xi } \le C\left( {Q,p,s,\gamma} \right).
\end{equation}
Applying Proposition \ref{Pro41} and \eqref{eq487} once again, we obtain
\begin{equation}\label{eq488}
\mathop {\sup }\limits_{0 < \left| h \right| < {h_0}} \int_{{B_{{R_1} - 4{h_0}}}} {{{\left| {\frac{{\Delta _{Z,h}^2u}}{{{{\left| h \right|}^{\frac{1}{{{q_2}}} + 1}}}}} \right|}^{{q_2}}}d\xi } \le c(1 + \Lambda )\left( {\int_{{B_{{{{R_1} + 4{h_0}}}}}} {{{\left| {{\nabla _H}u} \right|}^{{q_1}}}d\xi } + 1} \right) \le C\left( {Q,p,s,\gamma} \right).
\end{equation}
Moreover, by Lemma \ref{Le213} in \eqref{eq488}, we have
\begin{equation}\label{eq489}
\int_{{B_{{R_2} + 4{h_0}}}} {{{\left| {{\nabla _H}{u}} \right|}^{{q_2}}}d\xi } \le C\left( {Q,p,s,\gamma} \right).
\end{equation}
Repeating this procedure, it yields
\begin{equation}\label{eq490}
 \int_{{B_{{R_{i+1}} + 4{h_0}}}} {{{\left| {{\nabla _H}{u}} \right|}^{{q_{i+1}}}}d\xi } \le C\left( {Q,p,s,\gamma} \right)
\end{equation}
for all $i = 0, \cdots ,{i_\infty }-1.$ Taking $i={i_\infty }-1$ in \eqref{eq490} and using ${\left\| {{u}} \right\|_{{L^\infty }\left( {{B_1}} \right)}} \le 1$, we deduce
\begin{equation}\label{eq491}
{\left\| u \right\|_{H{W^{1,{q_{{i_\infty }}}}}\left( {{B_{{R_{{i_\infty }}} + 4{h_0}}}} \right)}} \le C\left( {Q,p,s,\gamma } \right).
\end{equation}
By means of \eqref{eq492}, we know ${q_{{i_\infty }}}>Q$, so we use ${R_{{i_\infty }}} + 4{h_0} = \frac{3}{4}$ and Lemma \ref{Le23} to get
$u \in C_{loc}^\gamma \left( {{B_{\frac{3}{4}}}} \right)$. In fact, we take a cut-off function $\psi$ between ${{B_{\frac{1}{2}}}}$ and ${{B_{\frac{3}{4}}}}$. Then
\begin{align}\label{eq493}
{\left\| u \right\|_{{C^\gamma }\left( {{B_{\frac{1}{2}}}} \right)}}& \le {\left\| {u\psi } \right\|_{{C^\gamma }\left( {{{\mathbb{H}}^n}} \right)}} \le c{\left\| {u\psi } \right\|_{H{W^{1,{q_{{i_\infty }}}}}\left( {{{\mathbb{H}}^n}} \right)}} = c{\left\| {u\psi } \right\|_{H{W^{1,{q_{{i_\infty }}}}}\left( {{B_{\frac{3}{4}}}} \right)}}\nonumber\\
& \le c{\left\| u \right\|_{H{W^{1,{q_{{i_\infty }}}}}\left( {{B_{\frac{3}{4}}}} \right)}} \le C\left( {Q,p,s,\gamma } \right).
\end{align}
Therefore, Theorem is proved.

\textbf{Proof of Theorem \ref{Th14}} After rescaling the variables as in the proof of Theorem \ref{Th13}, it suffices to show that
$${\left\| u_R \right\|_{{C^{1,\alpha }}\left( {{B_{\frac{1}{8}}}} \right)}} \le c\left( {Q,p,s,\gamma } \right),$$
where the function $u_R$ defined in \eqref{eq484} satisfies the following conditions:
\begin{equation}\label{eq52}
{\left\| {{u_R}} \right\|_{{L^\infty }\left( {{B_1}} \right)}} \le 1,\;{\int _{{\mathbb{H}^n}\backslash {B_1}}}\frac{{{{\left| {{u_R}\left( \eta \right)} \right|}^{p - 1}}}}{{\left\| \eta \right\|_{{\mathbb{H}^n}}^{Q + sp}}}d\eta \le 1.
\end{equation}
By Theorem \ref{Th13}, there exists $\gamma>\frac{sp}{p-1}$ such that
\[\mathop {\sup }\limits_{\xi \ne \eta \in {B_{\frac{1}{2}}}\left( {{\xi _0}} \right)} \frac{{\left| {{u_R}\left( \xi \right) - {u_R}\left( \eta \right)} \right|}}{{{{\left\| {{\eta ^{ - 1}} \circ \xi } \right\|}_{{\mathbb{H}^n}}^\gamma }}} \le c\left( {Q,p,s,\gamma } \right).\]
Now take any $\xi_0 \in {B_{\frac{1}{4}}}$, then we deduce from the above formula and the choice of $\gamma$ that
\[\int_{{B_{\frac{1}{4}}}\left( {{\xi _0}} \right)} {\frac{{{{\left| {{u_R}\left( {{\xi _0}} \right) - {u_R}\left( \eta \right)} \right|}^{p - 1}}}}{{{{\left\| {{\eta ^{ - 1}} \circ {\xi _0}} \right\|}_{{\mathbb{H}^n}}^{Q + sp}}}}d\eta } \le c\left( {Q,p,s,\gamma } \right)\int_{{B_{\frac{1}{4}}}\left( {{\xi _0}} \right)} {\frac{1}{{{{\left\| {{\eta ^{ - 1}} \circ {\xi _0}} \right\|}_{{\mathbb{H}^n}}^{Q + sp - \gamma \left( {p - 1} \right)}}}}d\eta } = c\left( {Q,p,s,\gamma } \right).\]
Moreover, we have by \eqref{eq52}
\[\int_{{\mathbb{H}^n}\backslash {B_{\frac{1}{4}}}\left( {{\xi _0}} \right)} {\frac{{{{\left| {{u_R}\left( {{\xi _0}} \right) - {u_R}\left( \eta \right)} \right|}^{p - 1}}}}{{{{\left\| {{\eta ^{ - 1}} \circ {\xi _0}} \right\|}_{{\mathbb{H}^n}}^{Q + sp}}}}d\eta } \le c\left( {Q,p,s} \right).\]
Hence
\[{\left\| {{{{{\left( { - \Delta _{{{\mathbb{H}},p}}} \right)}^s}}}{u_R}} \right\|_{{L^\infty }\left( {{B_{\frac{1}{4}}}} \right)}} \le c\left( {Q,p,s,\gamma } \right),\]
and also
\[{\left\| {{{{\left( { - \Delta _{{{\mathbb{H}},p}}} \right)}}}{u_R}} \right\|_{{L^\infty }\left( {{B_{\frac{1}{4}}}} \right)}} \le c\left( {Q,p,s,\gamma } \right),\]
which together with \eqref{eq52} and Theorem 1.2 in \cite{MZ21} imply
\[{\left\| {{u_R}} \right\|_{{C^{1,,\alpha }}\left( {{B_{\frac{1}{8}}}} \right)}} \le c\left( {Q,p,s,\gamma } \right).\]

\begin{remark}
In contrast to \cite{ZN26}, where the H\"older index $\gamma \in (0,\frac{sp}{p-1})$ relies on the parameters $s$ and $p$, Theorem \ref{Th13} establishes a parameter-independent H\"older regularity with $\gamma \in (0,1)$. This holds under more general nonlinear frameworks and is particularly advantageous for subsequent regularity analysis.
\end{remark}

\begin{theorem}\label{Th14}
Let $2\le p <\infty,\;0<s<1,\;sp<p-1$ and $0\le \Lambda \le 1$. Suppose $\Omega \in \mathbb{H}^n$ is a bounded open set and $u \in HW_{{\rm{loc}}}^{1,p}\left( \Omega \right) \cap{ L_{sp}^{p - 1}\left( {{\mathbb{H}^n}} \right)} $ is a weak solution of \eqref{eq0}. Then $u \in C_{loc}^{1,\alpha} \left( \Omega  \right)$ for some $0 < \alpha  < 1$. Moreover, for some $0 < \alpha  < 1$ and any Kor\'{a}nyi ball ${B_{2R}}\left( {{\xi _0}} \right) \subset  \subset \Omega $ with $0<R<1$, there exists a positive constant $c=c(Q,p,q,s,\gamma)$ such that
\begin{equation}\label{eq51}
 \mathop {\sup }\limits_{\xi  \ne \eta  \in {B_{\frac{R}{8}}}\left( {{\xi _0}} \right)} \frac{{\left| {{\nabla _H}u\left( \xi  \right) - {\nabla _H}u\left( \eta  \right)} \right|}}{{{{\left\| {{\eta ^{ - 1}} \circ \xi } \right\|}^\alpha _{\mathbb{H}^n}}}} \le \frac{c}{{{R^{1 + \alpha }}}}\left( {{{\left\| u \right\|}_{{L^\infty }\left( {{B_R}\left( {{\xi _0}} \right)} \right)}} + {\rm{Tai}}{{\rm{l}}_{p - 1,sp,p}}\left( {u;{\xi _0},R} \right)} \right),
\end{equation}
where the definition of ${\rm{Tai}}{{\rm{l}}_{p - 1,sp,p}}\left( {u;{\xi _0},R} \right)$ see \eqref{eq26} below.
\end{theorem}

The paper is organized as follows: In Section 2, we introduce relevant knowledge of the Heisenberg group, function spaces, and several necessary and new lemmas. In Section 3, we employ horizontal difference, along with certain difference estimates and the fractional Sobolev-type inequality on the Heisenberg group, to establish the crucial iteration scheme of Morrey-type for weak solutions to \eqref{eq0} required for the proof of the Theorem \ref{Th13}, i.e. Proposition \ref{Pro41}. In Section 4, we first establish a proposition regarding the gradient integrability of weak solutions by using Caccioppoli-type inequality, i.e. Proposition \ref{Pro42}. Then, by combining local boundedness, the Proposition \ref{Pro41}, the Proposition \ref{Pro42}, an iterative method, and the Morrey inequality, we prove Theorem \ref{Th13}. Finally, Theorem \ref{Th14} is derived by utilizing Theorem \ref{Th13} and Theorem 1.2 from \cite{MZ21}.

In the sequel, we denote by $c$ a generic positive constant which may vary from line to line. Relevant dependencies on parameters shall be emphasised utilizing parentheses, i.e., $c= c(n, p, q)$ means that $c$ depends on $n, p, q$.

To prove Theorem~\ref{Th13}, the paper proceeds as follows. In Section~3, it uses “horizontal difference, along with certain difference estimates and the fractional Sobolev-type inequality on the Heisenberg group” to “establish the crucial iteration scheme of Morrey-type for weak solutions to \eqref{eq0} required for the proof of the Theorem~\ref{Th13}, i.e. Proposition~\ref{Pro41}.” In Section~4, it “first establish[es] a proposition regarding the gradient integrability of weak solutions by using Caccioppoli-type inequality, i.e. Proposition~\ref{Pro42}.” Then, “by combining local boundedness, the Proposition~\ref{Pro41}, the Proposition~\ref{Pro42}, an iterative method, and the Morrey inequality, we prove Theorem~\ref{Th13}.”

To prove the main theorem, the paper proceeds as follows. Next, it uses “horizontal difference, along with certain difference estimates and the fractional Sobolev-type inequality on the Heisenberg group” to “establish the crucial iteration scheme of Morrey-type for weak solutions to
\begin{equation}\label{eq0}
 - \Delta _{{{\mathbb{H}},p}}u + \Lambda{\left( { - \Delta _{{{\mathbb{H}},p}}} \right)^s}u = 0\;{\rm{in}}\;\Omega ,
\end{equation}
required for the proof of the main theorem, i.e. the following proposition.

\begin{proposition}\label{Pro41}
Let $2\le p <\infty,\;0<s<1$ and $0\le \Lambda \le 1$. Assume $u \in HW_{{\rm{loc}}}^{1,p}\left( {{B_2}\left( {{\xi _0}} \right) \cap L_{sp}^{p - 1}\left( {{\mathbb{H}^n}} \right)} \right)$ is a weak solution of \eqref{eq0} in ${{B_2}\left( {{\xi _0}} \right)}$ satisfying
\begin{equation}\label{eq41}
{\left\| u \right\|_{{L^\infty }\left( {{B_1}\left( {{\xi _0}} \right)} \right)}} \le 1,\int_{{\mathbb{H}^n}\backslash {B_1}\left( {{\xi _0}} \right)} {\frac{{{{\left| {u\left( \eta  \right)} \right|}^{p - 1}}}}{{\left\| \eta  \right\|_{{\mathbb{H}^n}}^{Q + sp}}}d\eta }  \le 1.
\end{equation}
If $0<h_0<\frac{1}{10}$ and $R$ satisfy $4{h_0}<R\le 1-5{h_0}$, and ${\nabla _H}u \in {L^q}\left( {{B_{R + 4{h_0}}}\left( {{\xi _0}} \right)} \right)$ for some $q \ge p$, then
\begin{equation}\label{eq42}
 \mathop {\sup }\limits_{0 < \left| h \right| < {h_0}} \int_{{B_{R - {4h_0}}}} {{{\left| {\frac{{\Delta _{Z,h}^2u}}{{{{\left| h \right|}^{\frac{1}{{q + 1}} + 1}}}}} \right|}^{q + 1}}d\xi } \le c(1+\Lambda) \left(  {\int_{{B_{R + {4h_0}}}} {{{\left| {{\nabla _H}u} \right|}^q}d\xi } + 1} \right),
\end{equation}
where $c=c(Q,h_0,p,q,s)>0$.
\end{proposition}

Next, it “first establish[es] a proposition regarding the gradient integrability of weak solutions by using Caccioppoli-type inequality, i.e.” the following.

\begin{proposition}\label{Pro42}
Let $2\le p <\infty,\;0<s<1$ and $0\le \Lambda \le 1$. If $u \in HW_{{\rm{loc}}}^{1,p}\left( {{B_2}\left( {{\xi _0}} \right) \cap L_{sp}^{p - 1}\left( {{\mathbb{H}^n}} \right)} \right)$ is a weak solution of \eqref{eq0} in ${{B_{2}}\left( {{\xi _0}} \right)}$ satisfying \eqref{eq41}, then
\begin{equation*}
\int_{{B_{\frac{7}{8}}}\left( {{\xi _0}} \right)} {{{\left| {{\nabla _H}u} \right|}^p}d\xi }  \le C\left( {Q,p,s} \right).
\end{equation*}
\end{proposition}

Then, “by combining local boundedness, the proposition above, the preceding proposition, an iterative method, and the Morrey inequality, we prove the main theorem.”

\label{Th13}
Let $2\le p <\infty,\;0<s<1$ and $0\le \Lambda \le 1$. Suppose $\Omega \in \mathbb{H}^n$ is a bounded open set and $u \in HW_{{\rm{loc}}}^{1,p}\left( \Omega \right) \cap{ L_{sp}^{p - 1}\left( {{\mathbb{H}^n}} \right)} $ is a weak solution of
\begin{equation}\label{eq0}
 - \Delta _{{{\mathbb{H}},p}}u + \Lambda{\left( { - \Delta _{{{\mathbb{H}},p}}} \right)^s}u = 0\;{\rm{in}}\;\Omega ,
\end{equation}
Then $u \in C_{loc}^\gamma \left( \Omega  \right)$ for any $0 < \gamma  < 1$. Moreover, for any $0 < \gamma  < 1$ and any Kor\'{a}nyi ball ${B_{2R}}\left( {{\xi _0}} \right) \subset  \subset \Omega $ with $0<R<1$, there exists a positive constant $c=c(Q,p,s,\gamma)$ such that
\begin{equation}\label{eq482}
  \mathop {\sup }\limits_{\xi  \ne \eta  \in {B_{\frac{R}{2}}}\left( {{\xi _0}} \right)} \frac{{\left| {u\left( \xi  \right) - u\left( \eta  \right)} \right|}}{{{{\left\| {{\eta ^{ - 1}} \circ \xi } \right\|}^\gamma _{\mathbb{H}^n}}}} \le \frac{c}{{{R^\gamma }}}\left( {{{{\left\| u \right\|}_{{L^\infty }\left( {{B_R}\left( {{\xi _0}} \right)} \right)}}} + {\rm{Tai}}{{\rm{l}}_{p - 1,sp,p}}\left( {u;{\xi _0},R} \right)} \right),
\end{equation}
where ${\rm{Tai}}{{\rm{l}}_{p - 1,sp,p}}\left( {u;{\xi _0},R} \right)$ is given by
\begin{equation}\label{eq26}
 {\rm{Tai}}{{\rm{l}}_{q,\alpha ,\beta }}\left( {u;{\xi _0},R} \right) = {\left[ {{R^\beta }\int_{{{\mathbb{H}}^n}\backslash {B_R}\left( {{\xi _0}} \right)} {\frac{{{{\left| u \right|}^q}}}{{{{\left\| {\xi _0^{ - 1} \circ \xi } \right\|}^{Q + \alpha }_{\mathbb{H}^n}}}}d\xi } } \right]^{\frac{1}{q}}}.
\end{equation}

Question:
Let $2\le p<\infty$, $0<s<1$, and $0\le \Lambda\le 1$. Let $\Omega\subset \mathbb{H}^n$ be a bounded open set, let $Q=2n+2$, and suppose that
\[
u\in HW^{1,p}_{\mathrm{loc}}(\Omega)\cap L^{p-1}_{sp}(\mathbb{H}^n)
\]
is a weak solution in $\Omega$ of
\[
-\Delta_{\mathbb{H},p}u+\Lambda\,(-\Delta_{\mathbb{H},p})^s u=0,
\]
where
\[
-\Delta_{\mathbb{H},p}u=\operatorname{div}_H\big(|\nabla_H u|^{p-2}\nabla_H u\big)
\]
and
\[
(-\Delta_{\mathbb{H},p})^s u(\xi)=\mathrm{P.V.}\int_{\mathbb{H}^n}
\frac{|u(\xi)-u(\eta)|^{p-2}(u(\xi)-u(\eta))}{\|\eta^{-1}\circ \xi\|_{\mathbb{H}^n}^{Q+sp}}\,d\eta.
\]
For $q,\alpha,\beta>0$, define the nonlocal tail by
\[
\operatorname{Tail}_{q,\alpha,\beta}(u;\xi_0,R)=\left[R^\beta\int_{\mathbb{H}^n\setminus B_R(\xi_0)}
\frac{|u|^q}{\|\xi_0^{-1}\circ \xi\|_{\mathbb{H}^n}^{Q+\alpha}}\,d\xi\right]^{1/q}.
\]
Which of the following conclusions about $u$ holds under these assumptions?

$u\in C^\gamma_{\mathrm{loc}}(\Omega)$ for every $0<\gamma<1$. Moreover, for every $0<\gamma<1$ and every Kor\'anyi ball $B_{2R}(\xi_0)\Subset \Omega$ with $0<R<1$, there exists a constant $c=c(Q,p,s,\gamma)>0$ such that
\[
\sup_{\substack{\xi\ne \eta\\ \xi,\eta\in B_{R/2}(\xi_0)}}
\frac{|u(\xi)-u(\eta)|}{\|\eta^{-1}\circ \xi\|_{\mathbb{H}^n}^\gamma}
\le \frac{c}{R^\gamma}\left(\|u\|_{L^\infty(B_R(\xi_0))}+\operatorname{Tail}_{p-1,sp,p}(u;\xi_0,R)\right).
\]

Choices:
B.
$u\in C^\gamma_{\mathrm{loc}}(\Omega)$ for every $0<\gamma<\dfrac{sp}{p-1}$. Moreover, for every $0<\gamma<\dfrac{sp}{p-1}$ and every Kor\'anyi ball $B_{2R}(\xi_0)\Subset \Omega$ with $0<R<1$, there exists a constant $c=c(Q,p,s,\gamma)>0$ such that
\[
\sup_{\substack{\xi\ne \eta\\ \xi,\eta\in B_{R/2}(\xi_0)}}
\frac{|u(\xi)-u(\eta)|}{\|\eta^{-1}\circ \xi\|_{\mathbb{H}^n}^\gamma}
\le \frac{c}{R^\gamma}\left(\|u\|_{L^\infty(B_R(\xi_0))}+\operatorname{Tail}_{p-1,sp,p}(u;\xi_0,R)\right).
\]
C.
There exists some $\gamma\in(0,1)$ such that $u\in C^\gamma_{\mathrm{loc}}(\Omega)$. Moreover, for every such $\gamma$ and every Kor\'anyi ball $B_{2R}(\xi_0)\Subset \Omega$ with $0<R<1$, there exists a constant $c=c(Q,p,s,\gamma)>0$ such that
\[
\sup_{\substack{\xi\ne \eta\\ \xi,\eta\in B_{R/2}(\xi_0)}}
\frac{|u(\xi)-u(\eta)|}{\|\eta^{-1}\circ \xi\|_{\mathbb{H}^n}^\gamma}
\le \frac{c}{R^\gamma}\left(\|u\|_{L^\infty(B_R(\xi_0))}+\operatorname{Tail}_{p-1,sp,p}(u;\xi_0,R)\right).
\]
D.
$u\in C^\gamma_{\mathrm{loc}}(\Omega)$ for every $0<\gamma<1$. Moreover, for every $0<\gamma<1$ and every Kor\'anyi ball $B_{2R}(\xi_0)\Subset \Omega$ with $0<R<1$, there exists a constant $c=c(Q,p,s,\gamma)>0$ such that
\[
\sup_{\substack{\xi\ne \eta\\ \xi,\eta\in B_{R/2}(\xi_0)}}
\frac{|u(\xi)-u(\eta)|}{\|\eta^{-1}\circ \xi\|_{\mathbb{H}^n}^\gamma}
\le \frac{c}{R^\gamma}\left(\|u\|_{L^\infty(B_R(\xi_0))}+\operatorname{Tail}_{p-1,sp,p}(u;\xi_0,R)\right),
\]
where the constant $c$ can be chosen independently of $\gamma$.
E.
$u\in C^\gamma_{\mathrm{loc}}(\Omega)$ for every $0<\gamma<1$. Moreover, for every $0<\gamma<1$ and every Kor\'anyi ball $B_{2R}(\xi_0)\Subset \Omega$ with $0<R<1$, there exists a constant $c=c(Q,p,s,\gamma)>0$ such that
\[
\sup_{\substack{\xi\ne \eta\\ \xi,\eta\in B_R(\xi_0)}}
\frac{|u(\xi)-u(\eta)|}{\|\eta^{-1}\circ \xi\|_{\mathbb{H}^n}^\gamma}
\le \frac{c}{R^\gamma}\left(\|u\|_{L^\infty(B_R(\xi_0))}+\operatorname{Tail}_{p-1,sp,p}(u;\xi_0,R)\right).
\]