\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

The stem does not explicitly state the conclusion or encode the exact formula for the top Lyapunov exponent. It gives the hypotheses and asks which conclusion follows, without obvious lexical leakage toward choice A.

This is essentially a direct theorem-recall item: the hypotheses are laid out in full and the correct option is the standard conclusion for upper triangular random matrix cocycles. It is very close to a restatement rather than a problem requiring derived insight.

Some reasoning is needed to reject plausible alternatives such as the weaker inequality, the sum over diagonal terms, and the log-vs-expectation confusion. However, the item mainly tests recognition of the exact theorem rather than substantial generative reasoning.

The distractors are mathematically plausible and target realistic failure modes: confusing max with sum, accepting a merely weaker true statement, mixing pathwise and expectation-level claims, and swapping expectation of log with log of expectation.

A mathematically solid but theorem-recall-heavy MCQ. It avoids direct answer leakage and uses strong distractors, but it is close to a tautological restatement of a known result and only moderately pressures genuine reasoning.


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?

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. \[\gamma_1=\max_{i\in[d]}\,\lim_{n\to\infty}\frac1n\log\mathbb E\big|\big(X_n\big)(i,i)\big|,\qquad [d]=\{1,\dots,d\}.\]

The stem does not explicitly reveal the correct formula. It gives the setup and asks for the universal conclusion, without directly stating the equality in choice A.

This is essentially a direct restatement of a known theorem: under the listed hypotheses on upper triangular random matrices, the top Lyapunov exponent equals the maximum expected log of the diagonal entries. The item mainly asks the test-taker to recognize that theorem.

There is some reasoning pressure because the distractors exploit common confusions involving Jensen-type swaps, weaker inequalities, and moving expectations across limits/maxima. However, the item still primarily rewards recall of the exact theorem rather than genuine derivation.

The distractors are plausible and mathematically meaningful: B confuses log of expectation with expectation of log, C is a weaker true statement, D swaps max and expectation improperly, and E introduces a subtle expectation/limit mismatch. These align well with common failure modes.

A solid MCQ with no answer leakage and strong distractors, but it is mostly theorem recognition rather than a non-tautological test of generative reasoning.


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

The stem only defines the projector-like maps p_m and normalized chains N_m(M); it does not state or strongly hint at the decomposition theorem itself. The correct answer is not leaked by wording in the prompt.

The item is very close to a theorem-recall question: it asks which exact decomposition statement holds. However, it is not a pure verbatim restatement, since the options vary important features such as uniqueness, strict index inequalities, and normalizedness of coefficients.

Some reasoning is required to distinguish the exact valid statement from nearby alternatives, especially between the weaker true existence statement and the stronger unique decomposition statement. Still, the question mainly tests recognition of the precise theorem rather than substantial derivation.

The distractors are mathematically plausible and target realistic errors: allowing repeated indices, dropping uniqueness, confusing normalized coefficients with arbitrary ones, or making coefficients independent of x. They are distinct and well aligned with common failure modes.

A solid theorem-identification MCQ with strong distractors and no answer leakage, but it leans more toward precise recall/recognition than deep generative reasoning.