[\Cref{thm: main theorem} \& \Cref{thm: so do Lp}]
Let $X$ be a metric space with bounded geometry. If $X$ admits a coarse embedding into some $\ell^p$-space (or $L^p$-space) with $p\in[1,\infty)$, then for any invariant open subset $U$, the inclusion $i:\mathcal I(X,U)\to\mathcal G(X,U)$ induces an isomorphism on $K$-theory, i.e.,
$$i_*: K_*(\mathcal I(X,U))\to K_*(\mathcal G(X,U))$$
is an isomorphism.

\begin{Def}\label{def: Roe algebra}
The \emph{algebra Roe algebra} of $X$, denoted by $\IC[X,\H_X]$ (or $\IC[X]$ for simplicity), is a subalgebra of $\B(\H_X)$ consisting of all operators $T$ satisfying:
\begin{itemize}
\item $T$ has \emph{finite propagation} (or \emph{controlled propagation}), i.e., the \emph{support} of $T$
$$\supp(T)=\{(x,y)\in X\times X\mid T_{xy}\ne 0\},$$
is an entourage.
\item $T_{xy}\in\K(\H)$ for any $(x,y)\in X\times X$.
\end{itemize}
The completion of $\IC[X]$ in $\B(\H_X)$ is called the \emph{Roe algebra} of $X$, denoted by $C^*(X,\H_X)$ (or $C^*(X)$ for simplicity).
\end{Def}

\begin{Cor}\label{cor: CElp implies ONLFin}
If $X$ admits a coarse embedding into $\ell^p$, then $X$ has $ONL_{\P_{Fin}}$.
\end{Cor}

\begin{Def}\label{def: geom vs ghost}
Let $U\subseteq \beta X$ be an invariant open set. The \emph{geometric ideal} of $C^*(X)$ with respect to $U$, denoted by $\I(X, U)$, is defined to be the closed ideal generated by
$$\II[X,U]=\left\{T\in\IC[X]\ \big|\ \overline{r(\supp_{\varepsilon}(T))}\subseteq U\right\}.$$
The \emph{ghostly ideal} of $C^*_u(X)$ with respect to $U$, denoted by $\I_\G(X,U)$, is defined to be
$$\G(X,U)=\left\{T\in C^*(X)\ \big|\ \overline{r(\supp_{\varepsilon}(T))}\subseteq U\right\}.$$
\end{Def}

Ghosts are among the most mysterious operators in the Roe algebra. Let $X$ be a metric space with bounded geometry and fixed base point $x_0$. We denote $C^*(X)$ the Roe algebra of $X$ (\Cref{def: Roe algebra}). An operator $T \in C^*(X)$ can be written as a $X$-by-$X$ matrix. Such $T$ is called a \emph{ghost} if its matrix entries vanish at infinity; that is, $T_{xy} \to 0$ as $x, y \to \infty$. However, despite this local vanishing behavior, the operator $T$ is not necessarily compact. In other words, while the entries fade locally, the operator norm $\|\chi_{K_R}T\chi_{K_R}\|$ (where $K_R = B(x_0, R)$) may not tend to zero as $R \to \infty$. Consequently, a ghost is an object that is elusive locally at infinity but remains capturable globally. The collection of all such operators forms a closed ideal in the Roe algebra, known as the \emph{ghostly ideal}. It is also widely known as the \emph{ghost ideal} in the literature. For a detailed discussion on the construction of ghost projections and the structural analysis of ghostly ideals, we refer the reader to \cite{CW2004, WYI2012, KLVZ2021, LSZ2023, WZ2023}.

Ghostly ideals play a crucial role in the study of the coarse Baum-Connes conjecture by providing counterexamples. Let $X$ be a metric space with bounded geometry. There is a coarse assembly map given by
$$\mu: \lim_{d\to\infty}K_*(P_d(X))\to K_*(C^*(X)).$$
The coarse Baum-Connes conjecture states that this map is an isomorphism. However, if $X$ is a sequence of expander graphs, there exists a ghost projection in the Roe algebra $C^*(X)$. This projection defines an element in $K_0(C^*(X))$ that lies outside the image of the assembly map $\mu$, as shown in \cite{HLS2002, WYI2012}.

On the other hand, the study of ideal structures in Roe algebras is also fundamental for $K$-theoretic computations. In \cite{HRY1993}, N.~Higson, J.~Roe, and G.~Yu initially introduced geometric ideals associated with subspaces. They utilized this structure to establish the coarse Mayer-Vietoris principle, which allows for the reduction of $K$-theory calculations to subspaces. In \cite{CW2004}, X.~Chen and Q.~Wang extended the concept of geometric ideals from subspaces to invariant open sets in the coarse groupoid (see \Cref{def: geom vs ghost}). They proved that every ideal in the uniform Roe algebra corresponds to an invariant open subset of the coarse groupoid. Furthermore, they proved that if the metric space $X$ has Yu's Property A, then every ideal in the uniform Roe algebra is geometric. Later in \cite{RW2014}, J. Roe and R. Willett proved the converse, establishing that every ideal of the uniform Roe algebra is geometric if and only if the space has Property A.

More recently, Q. Wang and J. Zhang \cite{WZ2023} introduced a refined concept of ghostly ideals associated with specific invariant open sets (see \Cref{def: geom vs ghost}). Roughly speaking, for an invariant open set $U$, the corresponding ghostly ideal consists of operators whose matrix entries $T_{xy}$ vanish as $(x,y)$ tends to infinity along the direction of the complement $U^c$, rather than requiring them to vanish in all directions at infinity. When the subspace is bounded, the corresponding geometric ideal coincides exactly with the algebra of compact operators, and the corresponding ghostly ideal coincides with the original ghostly ideal of all ghost operators.

Although the requirement for these two ideals to be isomorphic at the algebraic level is very stringent (specifically, Property A), the conditions for their isomorphism at the $K$-theory level are much less restrictive. In \cite{FS2014}, M.~Finn-Sell proved that if a space $X$ admits a coarse embedding into a Hilbert space, the inclusion map from the compact operator algebra to the ghost ideal induces an isomorphism on $K$-theory. In \cite{WZ2023, WFZ2025}, this result was generalized to the embedding from general geometric ideals to ghostly ideals.

The coarse Baum-Connes conjecture is closely related to coarse embeddings into Hilbert spaces. G.~Yu's milestone result proved that if a metric space with bounded geometry admits a coarse embedding into Hilbert space, then the coarse Baum-Connes conjecture holds for it. This condition implies the a-T-menability of the coarse groupoid $G(X)$, see \cite{STY2002}. Indeed, previous results regarding the $K$-theory isomorphism between geometric and ghostly ideals rely heavily on the a-T-menability of the coarse groupoid.

Recently, the connection between the coarse Baum-Connes conjecture and the geometry of $\ell^p$-spaces ($1\le p<\infty$) has become increasingly tight. In \cite{WXYZ2024}, it proves that if a space coarsely embeds into an $\ell^p$-space, then the coarse Baum-Connes conjecture holds for $X$. Furthermore, it is shown that the coarse Novikov conjecture holds if $X$ admits a fibred coarse embedding into an $\ell^p$-space in \cite{GLWZ2024}. A natural question arises: does the $K$-theory isomorphism between geometric and ghostly ideals persist for spaces coarsely embeddable into $\ell^p$-spaces? In this paper, we prove the following result:

Ghosts are among the most mysterious operators in the Roe algebra. Let $X$ be a metric space with bounded geometry and fixed base point $x_0$. We denote $C^*(X)$ the Roe algebra of $X$ (\Cref{def: Roe algebra}). An operator $T \in C^*(X)$ can be written as a $X$-by-$X$ matrix. Such $T$ is called a \emph{ghost} if its matrix entries vanish at infinity; that is, $T_{xy} \to 0$ as $x, y \to \infty$. However, despite this local vanishing behavior, the operator $T$ is not necessarily compact. In other words, while the entries fade locally, the operator norm $\|\chi_{K_R}T\chi_{K_R}\|$ (where $K_R = B(x_0, R)$) may not tend to zero as $R \to \infty$. Consequently, a ghost is an object that is elusive locally at infinity but remains capturable globally. The collection of all such operators forms a closed ideal in the Roe algebra, known as the \emph{ghostly ideal}. It is also widely known as the \emph{ghost ideal} in the literature. For a detailed discussion on the construction of ghost projections and the structural analysis of ghostly ideals, we refer the reader to \cite{CW2004, WYI2012, KLVZ2021, LSZ2023, WZ2023}.

Ghostly ideals play a crucial role in the study of the coarse Baum-Connes conjecture by providing counterexamples. Let $X$ be a metric space with bounded geometry. There is a coarse assembly map given by
$$\mu: \lim_{d\to\infty}K_*(P_d(X))\to K_*(C^*(X)).$$
The coarse Baum-Connes conjecture states that this map is an isomorphism. However, if $X$ is a sequence of expander graphs, there exists a ghost projection in the Roe algebra $C^*(X)$. This projection defines an element in $K_0(C^*(X))$ that lies outside the image of the assembly map $\mu$, as shown in \cite{HLS2002, WYI2012}.

On the other hand, the study of ideal structures in Roe algebras is also fundamental for $K$-theoretic computations. In \cite{HRY1993}, N.~Higson, J.~Roe, and G.~Yu initially introduced geometric ideals associated with subspaces. They utilized this structure to establish the coarse Mayer-Vietoris principle, which allows for the reduction of $K$-theory calculations to subspaces. In \cite{CW2004}, X.~Chen and Q.~Wang extended the concept of geometric ideals from subspaces to invariant open sets in the coarse groupoid (see \Cref{def: geom vs ghost}). They proved that every ideal in the uniform Roe algebra corresponds to an invariant open subset of the coarse groupoid. Furthermore, they proved that if the metric space $X$ has Yu's Property A, then every ideal in the uniform Roe algebra is geometric. Later in \cite{RW2014}, J. Roe and R. Willett proved the converse, establishing that every ideal of the uniform Roe algebra is geometric if and only if the space has Property A.

Although the requirement for these two ideals to be isomorphic at the algebraic level is very stringent (specifically, Property A), the conditions for their isomorphism at the $K$-theory level are much less restrictive. In \cite{FS2014}, M.~Finn-Sell proved that if a space $X$ admits a coarse embedding into a Hilbert space, the inclusion map from the compact operator algebra to the ghost ideal induces an isomorphism on $K$-theory. In \cite{WZ2023, WFZ2025}, this result was generalized to the embedding from general geometric ideals to ghostly ideals.

The coarse Baum-Connes conjecture is closely related to coarse embeddings into Hilbert spaces. G.~Yu's milestone result proved that if a metric space with bounded geometry admits a coarse embedding into Hilbert space, then the coarse Baum-Connes conjecture holds for it. This condition implies the a-T-menability of the coarse groupoid $G(X)$, see \cite{STY2002}. Indeed, previous results regarding the $K$-theory isomorphism between geometric and ghostly ideals rely heavily on the a-T-menability of the coarse groupoid.

Recently, the connection between the coarse Baum-Connes conjecture and the geometry of $\ell^p$-spaces ($1\le p<\infty$) has become increasingly tight. In \cite{WXYZ2024}, it proves that if a space coarsely embeds into an $\ell^p$-space, then the coarse Baum-Connes conjecture holds for $X$. Furthermore, it is shown that the coarse Novikov conjecture holds if $X$ admits a fibred coarse embedding into an $\ell^p$-space in \cite{GLWZ2024}. A natural question arises: does the $K$-theory isomorphism between geometric and ghostly ideals persist for spaces coarsely embeddable into $\ell^p$-spaces? In this paper, we prove the following result:

\begin{abstract}
Let $X$ be a metric space with bounded geometry. We show that if $X$ admits a coarse embedding into an $\ell^p$-space ($1 \le p < \infty$), then the canonical inclusion from any geometric ideal to the corresponding ghostly ideal induces an isomorphism in $K$-theory. Our approach relies on the construction of twisted Roe algebras, avoiding the use of groupoid techniques. As consequences, we deduce the relative (maximal) coarse Baum-Connes conjectures for such spaces, as well as the Operator Norm Localization property for finite rank projections ($ONL_{\mathcal{P}_{Fin}}$).
\end{abstract}

Recently, the connection between the coarse Baum-Connes conjecture and the geometry of $\ell^p$-spaces ($1\le p<\infty$) has become increasingly tight. In \cite{WXYZ2024}, it proves that if a space coarsely embeds into an $\ell^p$-space, then the coarse Baum-Connes conjecture holds for $X$. Furthermore, it is shown that the coarse Novikov conjecture holds if $X$ admits a fibred coarse embedding into an $\ell^p$-space in \cite{GLWZ2024}. A natural question arises: does the $K$-theory isomorphism between geometric and ghostly ideals persist for spaces coarsely embeddable into $\ell^p$-spaces? In this paper, we prove the following result:

We emphasize that a-T-menability is linked to Hilbert space geometry. For a space $X$ that coarsely embeds into an $\ell^p$-space, its coarse groupoid is not necessarily a-T-menable. Consequently, our approach differs essentially from the methods used in \cite{WZ2023, WFZ2025}. Inspired by \cite{WXYZ2024}, we employ the Dirac-dual-Dirac construction to reduce the problem to the study of ideals in twisted algebras. At the level of twisted algebras, the structure readily implies the $K$-theory isomorphism between these two ideals.

\begin{Cor}[\Cref{cor: maximal corollary}]
Let $X$ be a metric space with bounded geometry that admits a coarse embedding into an $\ell^p$-space ($p\geq 1$). \begin{itemize}
\item[(1)] The canonical quotient map $\pi: C^*_{\max}(X)\to C^*(X)$ induces an isomorphism on $K$-theory, i.e.,
$$\pi_*: K_*(C^*_{\max}(X))\xrightarrow{\cong} K_*(C^*(X)).$$
\item[(2)] For any subspace $Y$, the canonical quotient map $\pi: C^*_{\max,Y,\infty}(X)\to C^*_{Y,\infty}(X)$ induces an isomorphism on the level of $K$-theory, i.e.,
$$\pi_*: K_*(C^*_{\max,Y,\infty}(X))\xrightarrow{\cong} K_*(C^*_{Y,\infty}(X)).$$
\item[(3)] For any subspace $Y$, the maximal relative coarse Baum-Connes conjecture holds for $(X, Y)$.
\end{itemize}\end{Cor}

\begin{Thm}\label{thm: main theorem}
Let $X$ be a metric space with bounded geometry.
If $X$ admits a coarse embedding into some $\ell^p$-space with $p\in[1,\infty)$, then for any invariant open subset $U$, the inclusion $i:\I(X,U)\to\G(X,U)$ induces an isomorphism on $K$-theory, i.e.,
$$i_*: K_*(\I(X,U))\to K_*(\G(X,U))$$
is an isomorphism.
\end{Thm}

For an invariant open subset $U \subseteq \beta X$, let $\I$ denote either the geometric ideal $\I(X,U)$ or the ghostly ideal $\G(X,U)$. By taking intersections with $\I$ in the pushout diagram of the geometric and ghostly ideals mentioned above, we obtain the following pushout diagram:
\[\begin{tikzcd}
\I(X,U_{Y\cap Z})\cap \I \arrow[d] \arrow[r] & \I(X,U_{Z})\cap \I \arrow[d] \\
\I(X,U_{Y})\cap \I \arrow[r] & \I
\end{tikzcd}\]
By the Five Lemma, it suffices to prove the canonical inclusion map
$$i:\I(X,U_{W})\cap \I(X,U)\to \I(X,U_{W})\cap \G(X,U)$$
induces an isomorphism on $K$-theory for $W=Y,Z$ and $Y\cap Z$ which are all sparse. By \Cref{lem: ideal and subalgebra cap I}, we conclude that $i_*$ is identified with
$$i_*: K_*(C^*(W)\cap \I(X,U))\to K_*(C^*(W)\cap \G(X,U)).$$
By \Cref{lem: ideal on subalgebra is still an ideal}, the map $i_*$ is further identified with
$$i_*: K_*(\I(W,U^{\beta W}))\to K_*(\G(W,U^{\beta W})).$$
Therefore, it suffices to prove \Cref{thm: main theorem} for the case when $X$ is a sparse space, as the general case will then follow. This completes the proof of the proposition.
\end{proof}

\begin{proof}[Proof of \Cref{cor: relative CBC}]
Consider the following exact sequence:
\[\begin{tikzcd}
{K_i(A_{Y,L,d})} \arrow[r] \arrow[d, "(1)"] & {K_i(A_{L,d})} \arrow[d, "(2)"] \arrow[r] & {K_i(A_{L,Y,\infty,d})} \arrow[d, "(3)"] \arrow[r] & {K_{i+1}(A_{Y,L,d})} \arrow[r] \arrow[d, "(4)"] & {K_{i+1}(A_{L, d})} \arrow[d, "(5)"] \\
{K_i(A_Y)} \arrow[r] & {K_i(A)} \arrow[r] & {K_i(A_{Y,\infty})} \arrow[r] & {K_{i+1}(A_Y)} \arrow[r] & {K_{i+1}(A)}
\end{tikzcd}\]
where $A_{Y,L,d} = C^*_{L,Y}(P_d(X))$, $A_{L,d} = C^*_{L}(P_d(X))$, $A_{L,Y,\infty,d} = C^*_{L,Y,\infty}(P_d(X))$, $A_Y = \G(X,U_Y)$, $A = C^*(X)$, and $A_{Y,\infty} = C^*_{Y,\infty}(X)$.
Since $X$ admits a coarse embedding into $\ell^p$, by \cite{WXYZ2024}, we have that the maps (2) and (5) are isomorphism for both $i=0,1$. By \Cref{thm: main theorem}, $K_*(\G(X,U_Y))\cong K_*(\I(X,U_Y))\cong K_*(C^*(Y))$. Thus, the maps (1) and (4) are identified with the coarse Baum-Connes assembly map for $Y$, which are also isomorphisms since $Y$ also admits a coarse embedding into $\ell^p$. Then the corollary follows from the Five Lemma.
\end{proof}

\begin{Cor}\label{cor: maximal corollary}
Let $X$ be a metric space with bounded geometry that admits a coarse embedding into an $\ell^p$-space ($p\geq 1$). \begin{itemize}
\item[(1)] The canonical quotient map $\pi: C^*_{\max}(X)\to C^*(X)$ induces an isomorphism on $K$-theory, i.e.,
$$\pi_*: K_*(C^*_{\max}(X))\xrightarrow{\cong} K_*(C^*(X)).$$
\item[(2)] For any subspace $Y$, the canonical quotient map $\pi: C^*_{\max,Y,\infty}(X)\to C^*_{Y,\infty}(X)$ induces an isomorphism on the level of $K$-theory, i.e.,
$$\pi_*: K_*(C^*_{\max,Y,\infty}(X))\xrightarrow{\cong} K_*(C^*_{Y,\infty}(X)).$$
\item[(3)] For any subspace $Y$, the maximal relative coarse Baum-Connes conjecture holds for $(X, Y)$.
\end{itemize}\end{Cor}

\begin{Cor}\label{cor: relative CBC}
Let $X$ be a metric space with bounded geometry that admits a coarse embedding into an $\ell^p$-space ($p\geq 1$). For any subspace $Y$, the relative coarse Baum-Connes conjecture holds for $(X, Y)$. In particular, the boundary coarse Baum-Connes conjecture holds for $X$.
\end{Cor}

\begin{Thm}\label{thm: so do Lp}
The conclusion of Theorem 2.6 holds true if the assumption of coarse embeddability into $\ell^p$ is replaced by coarse embeddability into a $L^p$-space ($1 \le p < \infty$).\qed
\end{Thm}

We emphasize that a-T-menability is linked to Hilbert space geometry. For a space $X$ that coarsely embeds into an $\ell^p$-space, its coarse groupoid is not necessarily a-T-menable. Consequently, our approach differs essentially from the methods used in \cite{WZ2023, WFZ2025}. Inspired by \cite{WXYZ2024}, we employ the Dirac-dual-Dirac construction to reduce the problem to the study of ideals in twisted algebras. At the level of twisted algebras, the structure readily implies the $K$-theory isomorphism between these two ideals.

As applications of our main result, we derive the following corollaries. The first corollary concerns the relative coarse Baum-Connes conjecture, introduced in \cite{GWZ2025}. Its validity characterizes local obstructions to positive scalar curvature at infinity for manifolds.

\begin{Cor}[\Cref{cor: relative CBC}]
Let $X$ be a metric space with bounded geometry that admits a coarse embedding into an $\ell^p$-space ($p\geq 1$). For any subspace $Y$, the relative coarse Baum-Connes conjecture holds for $(X, Y)$. \end{Cor}

The second concerns the operator norm localization for equi-approximable finite-rank projections, a concept introduced in \cite{BFV2024} that is of interest in geometric group theory. In this paper, we establish a connection between this condition and the geometry of $\ell^p$-spaces via $K$-theory computations.

\begin{Cor}[\Cref{cor: CElp implies ONLFin}]
Let $X$ be a metric space with bounded geometry. If $X$ admits a coarse embedding into $\ell^p$, then $X$ has operator norm localization for equi-approximable finite-rank projections.
\end{Cor}

Finally, we extend our results from the reduced setting to the maximal setting. By combining our main theorem with the techniques in \cite{HIT2020, WXYZ2024}, we deduce the following result regarding the maximal coarse Baum-Connes conjecture:

\begin{Cor}[\Cref{cor: maximal corollary}]
Let $X$ be a metric space with bounded geometry that admits a coarse embedding into an $\ell^p$-space ($p\geq 1$). \begin{itemize}
\item[(1)] The canonical quotient map $\pi: C^*_{\max}(X)\to C^*(X)$ induces an isomorphism on $K$-theory, i.e.,
$$\pi_*: K_*(C^*_{\max}(X))\xrightarrow{\cong} K_*(C^*(X)).$$
\item[(2)] For any subspace $Y$, the canonical quotient map $\pi: C^*_{\max,Y,\infty}(X)\to C^*_{Y,\infty}(X)$ induces an isomorphism on the level of $K$-theory, i.e.,
$$\pi_*: K_*(C^*_{\max,Y,\infty}(X))\xrightarrow{\cong} K_*(C^*_{Y,\infty}(X)).$$
\item[(3)] For any subspace $Y$, the maximal relative coarse Baum-Connes conjecture holds for $(X, Y)$.
\end{itemize}\end{Cor}

Our results naturally motivate further inquiries regarding more general coefficients and embedding conditions. We propose the following open questions:

\begin{Que}
In \cite{DG2024}, a twisted version of the coarse Baum-Connes conjecture is introduced. It is natural to ask whether the twisted coarse Baum-Connes conjecture holds for $X$ with any twisted coefficient algebra if $X$ admits a coarse embedding into $\ell^p$-spaces?

Furthermore, parallel with \cite{CWY2013}, does the maximal coarse Baum-Connes conjecture hold for $X$ if $X$ admits a fibred coarse embedding into $\ell^p$-space, with $p\in[1,\infty)$?
\end{Que}

The paper is organized as follows. In \Cref{sec: preliminary}, we recall some preliminaries on coarse geometry, and ideal structures of Roe algebras. \Cref{sec: proof} is devoted to the proof of our main theorem in four steps. Finally, in \Cref{sec: applications}, we discuss applications to the relative and maximal coarse Baum-Connes conjectures, as well as the operator norm localization property.

We “employ the Dirac-dual-Dirac construction to reduce the problem to the study of ideals in twisted algebras.” Since for $X$ coarsely embedding into an $\ell^p$-space “its coarse groupoid is not necessarily a-T-menable,” the approach “differs essentially” from a-T-menability methods. “At the level of twisted algebras, the structure readily implies the $K$-theory isomorphism between these two ideals.” The proof of the main theorem is then carried out “in four steps” (in \Cref{sec: proof}).

[\begin{Thm}\label{thm: main theorem}
Let $X$ be a metric space with bounded geometry.
If $X$ admits a coarse embedding into some $\ell^p$-space with $p\in[1,\infty)$, then for any invariant open subset $U$, the inclusion $i:\I(X,U)\to\G(X,U)$ induces an isomorphism on $K$-theory, i.e.,
$$i_*: K_*(\I(X,U))\to K_*(\G(X,U))$$
is an isomorphism.
\end{Thm} \& \begin{Thm}\label{thm: so do Lp}
The conclusion of Theorem 2.6 holds true if the assumption of coarse embeddability into $\ell^p$ is replaced by coarse embeddability into a $L^p$-space ($1 \le p < \infty$).\qed
\end{Thm}]
Let $X$ be a metric space with bounded geometry. If $X$ admits a coarse embedding into some $\ell^p$-space (or $L^p$-space) with $p\in[1,\infty)$, then for any invariant open subset $U$, the inclusion $i:\mathcal I(X,U)\to\mathcal G(X,U)$ induces an isomorphism on $K$-theory, i.e.,
$$i_*: K_*(\mathcal I(X,U))\to K_*(\mathcal G(X,U))$$
is an isomorphism.,

Question: Let $X$ be a metric space with bounded geometry, and assume that $X$ admits a coarse embedding into some $ell^p$-space or $L^p$-space for some $p\in[1,\infty)$. For an invariant open subset $U$ of the coarse groupoid of $X$, let $\mathcal I(X,U)$ denote the geometric ideal in the Roe algebra $C^*(X)$ associated to $U$, and let $\mathcal G(X,U)$ denote the corresponding ghostly ideal. Under these hypotheses, which statement about the canonical inclusion $i:\mathcal I(X,U)\to\mathcal G(X,U)$ holds?

For every invariant open subset $U$, the inclusion $i:\mathcal I(X,U)\to\mathcal G(X,U)$ induces an isomorphism on $K$-theory; equivalently, $$i_*:K_*(\mathcal I(X,U))\xrightarrow{\cong}K_*(\mathcal G(X,U))$$ is an isomorphism.

Choices:
B. For every invariant open subset $U$, the inclusion $i:\mathcal I(X,U)\to\mathcal G(X,U)$ is itself an isomorphism of $C^*$-algebras; in particular, $$i_*:K_*(\mathcal I(X,U))\xrightarrow{\cong}K_*(\mathcal G(X,U))$$ is an isomorphism.
C. There exists at least one invariant open subset $U$ for which the inclusion $i:\mathcal I(X,U)\to\mathcal G(X,U)$ induces an isomorphism on $K$-theory; that is, $$i_*:K_*(\mathcal I(X,U))\xrightarrow{\cong}K_*(\mathcal G(X,U))$$ for some $U$.
D. For every invariant open subset $U$, the inclusion $i:\mathcal I(X,U)\to\mathcal G(X,U)$ induces an injection on $K$-theory, i.e., $$i_*:K_*(\mathcal I(X,U))\hookrightarrow K_*(\mathcal G(X,U)),$$ but it is not asserted to be surjective in general.
E. If $X$ admits a coarse embedding into some $\ell^p$-space with $p\in[1,\infty)$, then for every invariant open subset $U$ the inclusion $i:\mathcal I(X,U)\to\mathcal G(X,U)$ induces an isomorphism on $K$-theory; however, no such conclusion is claimed when the coarse embedding is only into a $L^p$-space.

K-theory isomorphism upgraded to algebra isomorphism

universal quantifier over invariant open subsets

surjectivity part of the K-theory isomorphism

extension from $\ell^p$ to $L^p$ embeddings


\begin{theorem}\label{thm-Serrin}
Let \((M^n, g)\) be a complete, connected, and noncompact  Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature. Assume that \(f\in C(\mathbb R_+;\mathbb R_+)\) satisfies
\begin{equation}\label{cond-Serrin}
\int_0^1[f(v)]^{-\frac{1}{2_*(q)-1}}\mathrm dv<+\infty.
\end{equation}
Then any positive supersolution to \eqref{Meq-gradient} must be constant.
\end{theorem}

\label{sec-introduction}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature.
Consider a class of semilinear equations with a gradient term
\begin{equation}\label{Meq-gradient}
-\Delta u = f(u)|\nabla u|^q\quad\text{in }M^n,
\end{equation}
where \(\Delta\) is the Laplace-Beltrami operator, \(\nabla\) is the gradient operator, and \(f:\mathbb R_+\to\mathbb R_+\) is continuous.
The equation \eqref{Meq-gradient} has a strong physical background. Such as a qualitative mathematical model studying the groundwater flow in a water-absorbing fissured porous rock in one spatial dimension (see e.g., \cite{BBCP2000} ), the porous media equation (see e.g., \cite{P2018}).

When \(f=0\), equation \eqref{Meq-gradient} becomes \(\Delta u=0\). Cheng-Yau \cite{CY1975} proved that any positive or bounded solution from above or below to \eqref{Meq-gradient} is a constant on Riemannian manifolds with nonnegative Ricci curvature.

For \(q=0\), the equation \eqref{Meq-gradient} becomes
\begin{equation}\label{Meq-q-0}
-\Delta u =f(u)\quad\text{in }M^n.
\end{equation}
For more general nonlinearity \(f(x,u)\), with additional assumptions imposed on \(f\), Gidas-Spruck \cite[Theorem 6.1]{GS1981} proved that solutions of equation \eqref{Meq-q-0} must be constant. In the Euclidean case, Serrin-Zou \cite{SZ2002} studied the \(m\)-Laplacian case. By focusing on \(m=2\), they proposed the subcritical condition on \(f\), that is
\[f(u)\geqslant0,\quad (p-1)f(u)-uf'(u)\geqslant0\]
for \(u>0, 1<p<\frac{2n}{n-2}\). Under the subcritical condition on \(f\), they obtained a Liouville theorem of equation \eqref{Meq-q-0} in \(\mathbb R^n\), that is if one of following condition holds:

(i) If \(n=2, 3\) and \(f\) is subcritical,

(ii) If \(n\geqslant 4\), \(f(u)\geqslant Cu\) for sufficiently large \(u\), and \(f\) is subcritical,

\noindent then equation \eqref{Meq-q-0} admits only constant solutions. For more Liouville theorems involving equation \eqref{Meq-q-0}, see \cite{L2025, L2025-1, LZ2003, MW2024, Z2025} and the references therein.

A typical case for equation \eqref{Meq-q-0} is
\begin{equation}\label{Meq-q-0-ty}
-\Delta u =u^p\quad\text{in }M^n.
\end{equation}
This equation is the famous Lane-Emden equation, which is related to the Yamabe problem. Gidas-Spruck \cite{GS1981} used differential identities to derive a Liouville theorem, namely, there is no positive solution when \(1<p<\frac{n+2}{n-2}\) on a complete Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci tensor. Grigor’yan-Sun \cite{GS2014} proved that equation \eqref{Meq-q-0-ty} admits no positive solution under condition \(\operatorname{vol}(B_R(x_0))\leqslant CR^{\frac{2p}{p-1}}\ln^{\frac{1}{p-1}}R\), corresponding with the Serrin subcritical case \(p<\frac{n}{n-2}\) in dimension \(n\), when \(p>1\) on a connected geodesically complete Riemannian manifold.

In the critical case  \(p=\frac{n+2}{n-2}\), the equation \eqref{Meq-q-0-ty} becomes
\[-\Delta u =u^{\frac{n+2}{n-2}}\quad\text{in }M^n, \]
which is the Euler-Lagrange equation of the Sobolev inequality. The critical case in \(\mathbb R^n\) has been successfully resolved, as can be seen in \cite{ CGS1989, CL1991,GNN1981}, all positive solutions must be of the form
\begin{equation}\label{solution-q-0}
u(x)=\Big(\frac{\sqrt{n(n-2)}\lambda}{1+(\lambda|x-x_0|)^2}\Big)^{\frac{n-2}{2}}.
\end{equation}
On generally Riemannian manifolds, the classification result turns to the rigidity result: the manifold must be \(\mathbb R^n\) if there exists a nonconstant solution. However, deriving the rigidity result is more challenging. Fogagnolo-Malchiodi-Mazzieri \cite{FMM2023} got the rigidity result under the assumption \(u=O(r^{-\frac{n-2}{2}})\) at infinity. It is worth mentioning that Catino-Monticelli \cite{CM2022} obtained the rigidity result and classified it with \(n=3\), while \(n\geqslant4\) either under the finite energy condition or under the decay assumption of the solution at infinity. Ciraolo-Farina-Polvara \cite{CFP2024} extended these results to \(n=4,5\) without any finite energy condition by the \(P\)-function method. Catino-Monticelli-Roncoroni \cite{CMR2023}, Ou \cite{O2025}, and Sun-Wang \cite{SW2025} extended the classification results on manifolds to \(m\)-Laplacian case.

Now, we return to the case with the gradient term. A crucial prototype of semilinear equation \eqref{Meq-gradient} is the case \(f(u)=u^p\), i.e.,
\begin{equation}\label{eq-u^p}
-\Delta u = u^p|\nabla u|^q\quad\text{in }M^n.
\end{equation}
There are three important curves of \((p,q)\) for this equation, namely the homogeneous curve \(A_0(p,q)=0\), the first critical curve \(A_1(p,q)=0\), and the second critical curve \(A_2(p,q)=0\), with
\[A_0(p,q)=p+q-1,\quad A_1(p,q)=(n-2)p+(n-1)q-n,\]
\[A_2(p,q)=(n-2)p+(n-1)q-\Big(n+\frac{2-q}{1-q}\Big),\quad0\leqslant q<1.\]
As shown in Figure \ref{fig-1}, we mark these three curves with \(A_0\), \(A_1\), and \(A_2\) respectively. We introduce
\[2_*(q)=\frac{2-q}{n-2}(n-1),\quad 2^*(q)=\frac{2-q}{n-2}(n+\frac{q}{1-q}),\]
thus the first subcritical range \(A_1(p,q)<0\) is equivalent with \(p<2_*(q)-1\), and the second subcritical range \(A_2(p,q)<0\) is equivalent with \(p<2^*(q)-1\) when \(0\leqslant q<1\).

In the special case \(p=0\), equation \eqref{eq-u^p} becomes \(-\Delta u=|\nabla u|^q\), called as the Hamilton-Jacobi equation. For this equation, Lions \cite{L1985} obtained a Liouville theorem for \(q>1\) in \(\mathbb R^n\). Bidaut-V\'{e}ron, Huidobro, and V\'{e}ron \cite{BV-GH-V2014} established the gradient estimate to \(m\)-Laplacian case and obtained some Liouville theorems on complete noncompact manifolds, which satisfy a lower bound estimate on the Ricci curvature and sectional curvature.

In the Euclidean case, there has already been a great deal of work on equation \eqref{eq-u^p}. In the first subcritical case \(p<2_*(q)-1\), any supersolution to \eqref{eq-u^p} must be constant, see \cite{BP-GM-Q2016, BV-GH-V2019, CHZ2022, CM1997, F2009, MP2001}. Those past proofs are usually composed of three cases split by the homogeneous curve \(A_0(p,q)=0\), which is avoided in our proof of Theorem \ref{thm-Serrin}. Besides, for \(q\geqslant2\) and \(p\geqslant0\), Filippucci-Pucci-Souple \cite{FPS2020} obtained that any positive bounded solution to equation \eqref{eq-u^p} is constant. Later, Bidaut-V\'{e}ron \cite{BV2021} extended this result to the \(m\)-Laplace equation for \(q\geqslant m\) and established a Liouville theorem with \(p\geqslant 0\).

Now we concentrate on the second subcritical range \(p<2^*(q)-1\) with \(0\leqslant q<1\) for equation \eqref{eq-u^p} in \(\mathbb R^n\). For \(0<q<2\) and \(p>0\), Bidaut-V\'{e}ron, Huidobro, and V\'{e}ron \cite{BV-GH-V2019} conjectured that any solution to \eqref{eq-u^p} must be constant in the second critical range. They gave a positive answer to the left of the curve \(G(p,q)=0\) in Figure \ref{fig-1}, where
\[G(p,q)=\big((n-1)^2q+n-2\big)p^2+c(q)p-nq^2,\]
\[c(q)=n(n-1)q^2-(n^2+n-1)q-n-2.\]
Besides, they also showed that the second critical curve is optimal by proving the existence of a nonconstant solution in the second supercritical range \(A_2(p,q)>0\). Using Bernstein's technique, Chang-Hu-Zhang \cite{CHZ2022} derived a Liouville theorem in \(m\)-Laplacian case, and the left of the curve \(C\) in Figure \ref{fig-1} shows their range when \(m=2\). An exciting result was obtained by Ma-Wu \cite{MW2023}. With the help of integral identities, they completely solved the conjecture in \cite{BV-GH-V2019} when \(0<q<\frac{1}{n-1}\), and extended the range \(G(p,q)<0\) in \cite{BV-GH-V2019} to \(H(p,q)<0\) when \(q\geqslant\frac{1}{n-1}\), where
\[H(p,q)=p^2+\Big(\frac{n-1}{n-2}q-\frac{n^2-3}{(n-2)^2}\Big)p+\frac{1-(n-1)q}{(n-2)^2}.\]

\begin{figure}[htbp]\small
\centering
\includegraphics[width=0.8\textwidth]{images/diagram.jpg}
\caption{some parameter ranges of equation \eqref{eq-u^p} in \(\mathbb R^n\) when \(n=5\)}
\label{fig-1}
\end{figure}

Return to the Riemannian manifold case. Sun-Xiao-Xu \cite{SXX2022} obtained some Liouville theorems of equation \eqref{eq-u^p} under the growth of the geodesical ball and some range \((p,q)\in\mathbb R^2\) on a complete, noncompact Riemannian manifold. He-Hu-Wang \cite{HHW2023} obtained that any \(C^1\) smooth positive solution of equation \eqref{eq-u^p} is a constant when \(p+q<\frac{n+3}{n-1}\) on complete noncompact Riemannian manifold with nonnegative Ricci curvature. More generally, both \cite{HHW2023} and \cite{SXX2022} studied \(m\)-Laplacian case.

Thus, research on Liouville theorems for equation \eqref{eq-u^p} on manifolds is relatively scarce, let alone equation \eqref{Meq-gradient} with general semilinear term \(f(u)\), which motivates us to study equation \eqref{Meq-gradient} on manifolds. First, we consider supersolutions to \eqref{Meq-gradient}.

\begin{definition}
We say that \(u\) is a supersolution to \eqref{Meq-gradient} if \(u\in H_{\mathrm{loc}}^1(M^n)\cap C(M^n)\) satisfies
\begin{align}\label{weak-supsol}
\int_{M^n} \nabla u\cdot\nabla\varphi\geqslant\int_{M^n}f(u)|\nabla u|^q\varphi,\quad\varphi\in C_c^\infty(M^n),~\varphi\geqslant 0.
\end{align}
\end{definition}

\begin{remark}
By an approximation argument, the test function \(\varphi\) may be chosen from \(H^1(M^n)\) with compact support.
\end{remark}

Under an integrable condition on \(f\), we obtain the following Liouville theorem by Serrin's technique.

\label{sec-introduction}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature.
Consider a class of semilinear equations with a gradient term
\begin{equation}\label{Meq-gradient}
-\Delta u = f(u)|\nabla u|^q\quad\text{in }M^n,
\end{equation}
where \(\Delta\) is the Laplace-Beltrami operator, \(\nabla\) is the gradient operator, and \(f:\mathbb R_+\to\mathbb R_+\) is continuous.
The equation \eqref{Meq-gradient} has a strong physical background. Such as a qualitative mathematical model studying the groundwater flow in a water-absorbing fissured porous rock in one spatial dimension (see e.g., \cite{BBCP2000} ), the porous media equation (see e.g., \cite{P2018}).

A typical case for equation \eqref{Meq-q-0} is
\begin{equation}\label{Meq-q-0-ty}
-\Delta u =u^p\quad\text{in }M^n.
\end{equation}
This equation is the famous Lane-Emden equation, which is related to the Yamabe problem. Gidas-Spruck \cite{GS1981} used differential identities to derive a Liouville theorem, namely, there is no positive solution when \(1<p<\frac{n+2}{n-2}\) on a complete Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci tensor. Grigor’yan-Sun \cite{GS2014} proved that equation \eqref{Meq-q-0-ty} admits no positive solution under condition \(\operatorname{vol}(B_R(x_0))\leqslant CR^{\frac{2p}{p-1}}\ln^{\frac{1}{p-1}}R\), corresponding with the Serrin subcritical case \(p<\frac{n}{n-2}\) in dimension \(n\), when \(p>1\) on a connected geodesically complete Riemannian manifold.

In the critical case \(p=\frac{n+2}{n-2}\), the equation \eqref{Meq-q-0-ty} becomes
\[-\Delta u =u^{\frac{n+2}{n-2}}\quad\text{in }M^n, \]
which is the Euler-Lagrange equation of the Sobolev inequality. The critical case in \(\mathbb R^n\) has been successfully resolved, as can be seen in \cite{ CGS1989, CL1991,GNN1981}, all positive solutions must be of the form
\begin{equation}\label{solution-q-0}
u(x)=\Big(\frac{\sqrt{n(n-2)}\lambda}{1+(\lambda|x-x_0|)^2}\Big)^{\frac{n-2}{2}}.
\end{equation}
On generally Riemannian manifolds, the classification result turns to the rigidity result: the manifold must be \(\mathbb R^n\) if there exists a nonconstant solution. However, deriving the rigidity result is more challenging. Fogagnolo-Malchiodi-Mazzieri \cite{FMM2023} got the rigidity result under the assumption \(u=O(r^{-\frac{n-2}{2}})\) at infinity. It is worth mentioning that Catino-Monticelli \cite{CM2022} obtained the rigidity result and classified it with \(n=3\), while \(n\geqslant4\) either under the finite energy condition or under the decay assumption of the solution at infinity. Ciraolo-Farina-Polvara \cite{CFP2024} extended these results to \(n=4,5\) without any finite energy condition by the \(P\)-function method. Catino-Monticelli-Roncoroni \cite{CMR2023}, Ou \cite{O2025}, and Sun-Wang \cite{SW2025} extended the classification results on manifolds to \(m\)-Laplacian case.

Return to the Riemannian manifold case. Sun-Xiao-Xu \cite{SXX2022} obtained some Liouville theorems of equation \eqref{eq-u^p} under the growth of the geodesical ball and some range \((p,q)\in\mathbb R^2\) on a complete, noncompact Riemannian manifold. He-Hu-Wang \cite{HHW2023} obtained that any \(C^1\) smooth positive solution of equation \eqref{eq-u^p} is a constant when \(p+q<\frac{n+3}{n-1}\) on complete noncompact Riemannian manifold with nonnegative Ricci curvature. More generally, both \cite{HHW2023} and \cite{SXX2022} studied \(m\)-Laplacian case.

\begin{remark}
By an approximation argument, the test function \(\varphi\) may be chosen from \(H^1(M^n)\) with compact support.
\end{remark}

Under an integrable condition on \(f\), we obtain the following Liouville theorem by Serrin's technique.

\begin{equation}\label{Meq-q-0}
-\Delta u =f(u)\quad\text{in }M^n.
\end{equation}

\begin{equation}\label{eq-u^p}
-\Delta u = u^p|\nabla u|^q\quad\text{in }M^n.
\end{equation}

In both cases, \(\mathbf E=0\) holds in \(Z^c\) by letting \(R\to\infty\). For \(x\in\mathring Z\), \(\nabla u(x)=0\) yields \(\mathbf E=0\). By the continuity of \(\mathbf E\), we immediately conclude that \(\mathbf E=0\) on \(M\). Together with Lemma \ref{critical-thm-pf}, the proof is complete.
\qedsections/intro.tex0000664000000000000000000004363015111220774013270 0ustar rootroot\section{Introduction}\label{sec-introduction}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature.
Consider a class of semilinear equations with a gradient term
\begin{equation}\label{Meq-gradient}
-\Delta u = f(u)|\nabla u|^q\quad\text{in }M^n,
\end{equation}
where \(\Delta\) is the Laplace-Beltrami operator, \(\nabla\) is the gradient operator, and \(f:\mathbb R_+\to\mathbb R_+\) is continuous.
The equation \eqref{Meq-gradient} has a strong physical background. Such as a qualitative mathematical model studying the groundwater flow in a water-absorbing fissured porous rock in one spatial dimension (see e.g., \cite{BBCP2000} ), the porous media equation (see e.g., \cite{P2018}).

Under an integrable condition on \(f\), we obtain the following Liouville theorem by Serrin's technique.

When \(\liminf_{v\to0^+}f(v)>0\), the condition \eqref{cond-Serrin} trivially holds.

\begin{corollary}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature. Assume that \(f\in C(\mathbb R_+;\mathbb R_+)\) satisfies
\[\liminf_{v\to0^+}f(v)>0.\]
Then any positive supersolution to \eqref{Meq-gradient} must be constant.
\end{corollary}

\begin{corollary}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature. Then any positive supersolution to \eqref{eq-u^p} must be constant if \(p<2_*(q)-1\),
\end{corollary}

\begin{theorem}\label{thm-subcritical}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature, and \(0<q<\frac{1}{n-1}\). Assume that \(f\in C^1(\mathbb R_+;\mathbb R_+)\) satisfies the following conditions:
\begin{equation}\label{cond-sub-1}
f(u)\geqslant u^{\beta_0},~u>N,\quad\text{for some }\beta_0\in(1-q,2^*(q)-1)\text{ and }N>0,
\end{equation}
\begin{equation}
uf'(u)\leqslant\alpha_0 f(u),\quad\text{for some }\alpha_0\in[\beta_0,2^*(q)-1).
\end{equation}
Then any positive \(C^3\) solution to \eqref{Meq-gradient} must be constant if \(p<2^*(q)-1\).
\end{theorem}

\begin{corollary}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature, and \(0<q<\frac{1}{n-1}\). Then any positive \(C^3\) solution to \eqref{eq-u^p} must be constant.
\end{corollary}

\begin{theorem}\label{thm-critical}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\in\{3,4,5\}\) with nonnegative Ricci curvature, \(0<q<\frac{1}{n-1}\) and \(f(u)=u^{2^*(q)-1}\) in \eqref{Meq-gradient}. Let \(u\in C^3(M)\) be a positive solution to \eqref{Meq-gradient}, then either \(u\) is constant, or \((M^n, g)\) is isometric to \(\mathbb R^n\) and \(u\) is the form given by \eqref{solution}.
\end{theorem}

\begin{equation}\label{cond-Serrin}
\int_0^1[f(v)]^{-\frac{1}{2_*(q)-1}}\mathrm dv<+\infty.
\end{equation}

\begin{equation}\label{eq-u^p}
-\Delta u = u^p|\nabla u|^q\quad\text{in }M^n.
\end{equation}

\begin{equation}\label{solution}
U_{\lambda,x_0}(x):=\Big(\frac{(n+\frac{q}{1-q})^{\frac{1}{2-q}}(n-2)^{\frac{1-q}{2-q}}\lambda}{1+(\lambda|x-x_0|)^{\frac{2-q}{1-q}}}\Big)^{\frac{(n-2)(1-q)}{2-q}},\quad\lambda>0,~x_0\in\mathbb R^n.
\end{equation}

When \(\liminf_{v\to0^+}f(v)>0\), the condition \eqref{cond-Serrin} trivially holds.

\begin{corollary}
Let \((M^n, g)\) be a complete, connected, and noncompact  Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature. Assume that \(f\in C(\mathbb R_+;\mathbb R_+)\) satisfies
\[\liminf_{v\to0^+}f(v)>0.\]
Then any positive supersolution to \eqref{Meq-gradient} must be constant.
\end{corollary}

Theorem \ref{thm-Serrin} recovers the first subcritical range \(p<2_*(q)-1\) by taking \(f(u)=u^p\).

\begin{corollary}
Let \((M^n, g)\) be a complete, connected, and noncompact  Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature. Then any positive supersolution to \eqref{eq-u^p} must be constant if \(p<2_*(q)-1\),
\end{corollary}

For solutions to \eqref{Meq-gradient}, we concentrate on the cases \(0<q<\frac{1}{n-1}\). We establish the following Liouville theorem. Our proof also works for the case \(q=0\), which recovers the \(m=2\) case in Serrin-Zou \cite{SZ2002} in \(M=\mathbb R^n\).

\begin{theorem}\label{thm-subcritical}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature, and \(0<q<\frac{1}{n-1}\). Assume that \(f\in C^1(\mathbb R_+;\mathbb R_+)\) satisfies the following conditions:
\begin{equation}\label{cond-sub-1}
f(u)\geqslant u^{\beta_0},~u>N,\quad\text{for some }\beta_0\in(1-q,2^*(q)-1)\text{ and }N>0,
\end{equation}
\begin{equation}
uf'(u)\leqslant\alpha_0 f(u),\quad\text{for some }\alpha_0\in[\beta_0,2^*(q)-1).
\end{equation}
Then any positive \(C^3\) solution to \eqref{Meq-gradient} must be constant if \(p<2^*(q)-1\).
\end{theorem}

Theorem \ref{thm-subcritical} recovers the second subcritical range \(p<2^*(q)-1\) by taking \(f(u)=u^p\).

\begin{corollary}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature, and \(0<q<\frac{1}{n-1}\). Then any positive \(C^3\) solution to \eqref{eq-u^p} must be constant.
\end{corollary}

By taking \(M=\mathbb R^n\), this corollary recovers the case \(0<q<\frac{1}{n-1}\) in \cite[Theorem 1.3]{MW2023}, where Ma and Wu solved the conjectured raised in \cite{BV-GH-V2019}.

Lastly, we consider the second critical case \(f(u) = u^{2^*(q)-1}\). Bidaut-V\'{e}ron, Huidobro, and V\'{e}ron \cite[Theorem D]{BV-GH-V2019} found an explicit 1-parameter family of positive radial solutions in \(\mathbb R^n\)
\[u_c(r)=c(Kc^{\frac{(2-q)^2}{(n-2)(1-q)}}+r^{\frac{2-q}{1-q}})^{-\frac{(n-2)(1-q)}{2-q}},\quad c>0,~K=K(n,q)>0.\]
By inserting them into equation \eqref{Meq-gradient} with \(f(u) = u^{2^*(q)-1}\), we rewrite the solutions as
\begin{equation}\label{solution}
U_{\lambda,x_0}(x):=\Big(\frac{(n+\frac{q}{1-q})^{\frac{1}{2-q}}(n-2)^{\frac{1-q}{2-q}}\lambda}{1+(\lambda|x-x_0|)^{\frac{2-q}{1-q}}}\Big)^{\frac{(n-2)(1-q)}{2-q}},\quad\lambda>0,~x_0\in\mathbb R^n.
\end{equation}
For \(q=0\), the solution \eqref{solution} is equivalent to \eqref{solution-q-0}. The rigidity of manifolds has been rigorously established in low-dimensional settings for \(q=0\), but for \(0<q<1\), the classification of equation \eqref{Meq-gradient} in the second critical case \(f(u) = u^{2^*(q)-1}\) on \(\mathbb R^n\) is open, let alone the rigidity of manifolds.

With the help of the invariant tensor \(\operatorname{\mathbf E}\) and its corresponding pseudo-invariant function \(l\) (i.e., \(\nabla l\) is only composed of terms involving the invariant tensor \(\operatorname{\mathbf E}\)), we obtain the rigidity for the ambient manifold and classify positive solutions of the second critical equation, \eqref{Meq-gradient} with \(f(u)=u^{2^*(q)-1}\), without any assumption in a low-dimensional setting. To our knowledge, this is the first rigidity result in the second critical case of equations with gradient terms.

\begin{theorem}\label{thm-critical}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\in\{3,4,5\}\) with nonnegative Ricci curvature, \(0<q<\frac{1}{n-1}\) and \(f(u)=u^{2^*(q)-1}\) in \eqref{Meq-gradient}. Let \(u\in C^3(M)\) be a positive solution to \eqref{Meq-gradient}, then either \(u\) is constant, or \((M^n, g)\) is isometric to \(\mathbb R^n\) and \(u\) is the form given by \eqref{solution}.
\end{theorem}

Differential identities are crucial in studying properties of an elliptic equation. However, establishing differential identities usually relies on some geometric background, such as the Bochner formula. Thus, finding a proper differential identity may be challenging for a sophisticated equation. In \cite{MW2024}, Ma-Wu (the third author of this paper) developed the invariant tensor technique, a powerful method of finding vector fields and differential identities without any geometric instructions, to study semilinear equations \(-\Delta u=f(u)\) on \(M^n\), which gave a new proof of Bidaut-V\'{e}ron and V\'{e}ron's rigidity result in \cite{BV-V1991}. Using the new technique, Ma-Ou-Wu \cite{MOW2025} reconstructed Jerison-Lee's identity to answer the question raised by Jerison-Lee \cite{JL1988}. Besides, Ma-Ou-Wu \cite{MOW2023} verified the subcritical rigidity conjecture raised by Wang \cite{W2022} via this new technique with sophisticated computations. Recently, Guo-Zhang \cite{GZ2025} utilised the invariant tensor technique to give a new proof of semilinear equations again. The invariant tensor technique is also widely applied to classify quasilinear cases or involving gradient terms, see \cite{LWY2025, Z2024}.

Although we concentrate only on the case \(0<q<\frac{1}{n-1}\), we make the first attempt to successfully extend the invariant tensor technique to equations with gradient terms. The pseudo-invariant function \(l\), which plays a crucial role in the classification process, is found by differential invariance. However, it becomes difficult to find \(l\) through variable transformations as previously done. The authors have extended the invariant tensor technique to the equation \eqref{Meq-gradient} to \(m\)-Laplace in \cite{WZ}, where a wider range of \(q\) is studied with more sophisticated computations.

Throughout this paper, we assume constant \(R>1\), \(\varepsilon>0\) small enough, and \(C>0\) are constants independent of \(R\). Let \(B_R\) be the geodesic ball centered at some fixed point with radius \(R\). Unless otherwise specified, we employ the summation convention for repeated indices from \(1\) to \(n\). In proofs involving manifolds, we choose a local frame, \(\varphi_i\) denotes the covariant derivative of the function \(\varphi\). Let \(\eta\) be a smooth cutoff function supported in \(B_{2R}\) satisfying \(\eta\equiv 1\) in \(B_R\), \(0\leqslant\eta\leqslant 1\), and \(|\nabla\eta|\leqslant CR^{-1}\).

This paper is structured as follows. In Section 2, we prove Theorem \ref{thm-Serrin} by using Serrin's technique with a delicate test function \(b(u)\eta^n\) and a three-term Young's inequality. In Section 3, using the invariant tensor technique, we establish a key differential identity and necessary propositions to prove Theorem \ref{thm-subcritical}. With the help of this differential identity, the proof of Theorem \ref{thm-critical}, classification of solutions in the critical case, is finished in Section 4.sections/Serrin.tex0000664000000000000000000000411015111220774013365 0ustar  rootroot

To prove Theorem~\ref{thm-Serrin}, the paper uses “Serrin's technique with a delicate test function $b(u)\eta^n$ and a three-term Young's inequality” (carried out in Section 2).

To prove the main theorem, the paper uses “Serrin's technique with a delicate test function $b(u)\eta^n$ and a three-term Young's inequality” (carried out later).

\label{thm-Serrin}
Let \((M^n, g)\) be a complete, connected, and noncompact  Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature. Assume that \(f\in C(\mathbb R_+;\mathbb R_+)\) satisfies
\begin{equation}\label{cond-Serrin}
\int_0^1[f(v)]^{-\frac{1}{2_*(q)-1}}\mathrm dv<+\infty.
\end{equation}
Then any positive supersolution to \begin{equation}\label{Meq-gradient}
-\Delta u = f(u)|\nabla u|^q\quad\text{in }M^n,
\end{equation}
must be constant.,

Question:
Let \((M^n,g)\) be a complete, connected, noncompact Riemannian manifold of dimension \(n\ge 3\) with nonnegative Ricci curvature. Let \(f\in C(\mathbb R_+;\mathbb R_+)\) satisfy
\[
\int_0^1 [f(v)]^{-\frac{1}{2_*(q)-1}}\,dv<\infty,
\]
and consider the equation with gradient term
\[
-\Delta u=f(u)|\nabla u|^q\quad \text{in }M^n,
\]
where \(\Delta\) is the Laplace--Beltrami operator and \(\nabla u\) is the gradient of \(u\). If \(u\) is a positive supersolution, meaning \(u>0\) and \(-\Delta u\ge f(u)|\nabla u|^q\) on \(M^n\), which conclusion holds?

Every positive supersolution \(u\) is constant on \(M^n\).

Choices:
B. Every positive supersolution \(u\) is constant on \(M^n\) provided additionally that \(0<q<\frac{1}{n-1}\); without this restriction on \(q\), the conclusion need not hold.
C. Every bounded positive supersolution \(u\) is constant on \(M^n\).
D. There exists a constant \(C=C(M,n,q,f)>0\) such that every positive supersolution \(u\) satisfies \(u\equiv C\) on \(M^n\).
E. Every positive \(C^1\) solution \(u\) of \(-\Delta u=f(u)|\nabla u|^q\) on \(M^n\) is constant, but the same conclusion is not asserted for positive supersolutions satisfying only \(-\Delta u\ge f(u)|\nabla u|^q\).

no subcritical range restriction on q in the Serrin argument

dropped the full class of positive supersolutions to the smaller class of bounded positive supersolutions

constancy depends on the individual supersolution, not a universal constant determined only by the data

Serrin test-function method is stated for supersolutions, not merely for exact C^1 solutions


\label{sec: Introduction}

In the dynamics of entire function $f$, the complex plane can be divided into two sets: the Fatou set $F(f)$ and the Julia set $J(f)$, according to whether the iteration sequence forms a normal family locally. By the classification of Fatou components, the dynamics on $F(f)$ is clear. However, even for the quadratic polynomials, the dynamics on the Julia sets is quite complicated. By Carath\'eodory's theorem, if the boundary of a Fatou component $U$ is locally connected, then the dynamics of $f$ on $\partial U$ can be conjugate to a mapping of much simpler form. Therefore, the local connectivity of the boundaries of Fatou components and the Julia set is an important topic in complex dynamics.

The local connectivity of Julia sets has been studied for decades: Hyperbolic, subhyperbolic, semi-hyperbolic, geometrically finite and Collet-Eckmann polynomials always have locally connected Julia sets (\cite{DH84, CJY94, TY96, GS98}). These results have the analogy for transcendental entire functions (\cite{BM02, BFR15, Par22, ARS22}). Essentially, all these results assume that the mapping is expanding on a neighborhood of its Julia set. So a natural question is whether the Julia set (or the boundaries of Fatou components) is locally connected without the assumption on the expandness of $f$ on its Julia set.

In early 1990s, Yoccoz (\cite{Hub93}) introduced the puzzle technique for quadratic polynomials. With this technique, he was able to study the local connectivity of quadratic polynomials without assuming the expandness on the Julia sets. Later, his results were extended to all polynomials by Roesch and Yin (\cite{RY08}).

The puzzle technique is itself a powerful tool in the dynamics of polynomials. For example, it is involved in the study of the density of hyperbolicity in real polynomials (\cite{KSS07}) and the local connectivity of Julia sets of Newton mappings (\cite{Roe08}). The construction of puzzles in the polynomial cases is based on the fact that $\infty$ is a superattracting fixed point and the immediate basin of $\infty$ enjoys the structure of external rays and equipotentials. However, because of the absence of equipotentials, it becomes extremely difficult to construct Yoccoz puzzles for transcendental entire functions.

In this paper, we considered a simple family of transcendental entire functions, the cosine family $\{f(z)=ae^z+be^{-z}: a,b \in \mathbb{C} \setminus \{0\}\}$. With the puzzles, we obtained the local connectivity of the boundaries of Fatou components:

\label{sec: Introduction}

In the dynamics of entire function $f$, the complex plane can be divided into two sets: the Fatou set $F(f)$ and the Julia set $J(f)$, according to whether the iteration sequence forms a normal family locally. By the classification of Fatou components, the dynamics on $F(f)$ is clear. However, even for the quadratic polynomials, the dynamics on the Julia sets is quite complicated. By Carath\'eodory's theorem, if the boundary of a Fatou component $U$ is locally connected, then the dynamics of $f$ on $\partial U$ can be conjugate to a mapping of much simpler form. Therefore, the local connectivity of the boundaries of Fatou components and the Julia set is an important topic in complex dynamics.

The local connectivity of Julia sets has been studied for decades: Hyperbolic, subhyperbolic, semi-hyperbolic, geometrically finite and Collet-Eckmann polynomials always have locally connected Julia sets (\cite{DH84, CJY94, TY96, GS98}). These results have the analogy for transcendental entire functions (\cite{BM02, BFR15, Par22, ARS22}). Essentially, all these results assume that the mapping is expanding on a neighborhood of its Julia set. So a natural question is whether the Julia set (or the boundaries of Fatou components) is locally connected without the assumption on the expandness of $f$ on its Julia set.

The puzzle technique is itself a powerful tool in the dynamics of polynomials. For example, it is involved in the study of the density of hyperbolicity in real polynomials (\cite{KSS07}) and the local connectivity of Julia sets of Newton mappings (\cite{Roe08}). The construction of puzzles in the polynomial cases is based on the fact that $\infty$ is a superattracting fixed point and the immediate basin of $\infty$ enjoys the structure of external rays and equipotentials. However, because of the absence of equipotentials, it becomes extremely difficult to construct Yoccoz puzzles for transcendental entire functions.

In this paper, we considered a simple family of transcendental entire functions, the cosine family $\{f(z)=ae^z+be^{-z}: a,b \in \mathbb{C} \setminus \{0\}\}$. With the puzzles, we obtained the local connectivity of the boundaries of Fatou components:

The puzzle technique is itself a powerful tool in the dynamics of polynomials. For example, it is involved in the study of the density of hyperbolicity in real polynomials (\cite{KSS07}) and the local connectivity of Julia sets of Newton mappings (\cite{Roe08}). The construction of puzzles in the polynomial cases is based on the fact that $\infty$ is a superattracting fixed point and the immediate basin of $\infty$ enjoys the structure of external rays and equipotentials. However, because of the absence of equipotentials, it becomes extremely difficult to construct Yoccoz puzzles for transcendental entire functions.

It is worth noting that there are also some results about the local connectivity of the boundaries of Siegel disks. For example, for quadratic polynomials, if the rotation number of the Siegel disk is of bounded type, then the boundary of the disk is locally connected (\cite{Pet96}). And the local connectivity of Siegel disks holds for some families of transcendental entire functions (\cite{Zhang05,Yang13,Zhang16,ZFS20}).

The cosine function $f(z)=ae^{z}+be^{-z}$ has two critical values and no asymptotic values. So it shares some similarities with cubic polynomials. It is known that for a polynomial $p$ with at least one escaping critical orbit, or equivalently with $J(p)$ disconnected, $J(p)$ is a Cantor set if and only if the critical components of $K(p)$ is not periodic (\cite{BH92,QY09}). Here, $K(p)$, which is called the filled-in Julia set, is the complement of the immediate basin of $\infty$. In other words, if $p$ has periodic Fatou components and $J(p)$ is disconnected, then $p$ is renormalizable. We have the analogy for cosine functions:
\begin{theorem}\label{thm: cosine functions with an escaping critical value is renormalizable}
 Let $f$ be a cosine function. Suppose that the Fatou set is non-empty, and one of the critical values escapes to $\infty$. Then $f$ is renormalizable.
\end{theorem}

In this case, the local connectivity of Fatou components can be deduced easily.
\begin{corollary}\label{cor: Fatou components are locally connected in renormalizable case}
 Let $f$ be a cosine function. Suppose that the Fatou set is non-empty, and one of the critical values escapes to $\infty$. If $f$ has no Siegel disks, then all Fatou components of $f$ are Jordan domains.
\end{corollary}

Finally, with Theorem \ref{thm: bounded Fatou components are Jordan domains} and Corollary \ref{cor: Fatou components are locally connected in renormalizable case}, we conclude the local connectivity of $J(f)$:
\begin{theorem}\label{thm: local connectivity of Julia set}
 Let $f$ be a cosine function. Suppose that $F(f)$ is non-empty and has no Siegel disks. Suppose that a critical value $v$ lies in $F(f)$. Suppose that $f$ satisfies one of the following conditions:
 \begin{enumerate}
 \item the critical value $-v \in F(f)$ and is not in the same Fatou component as $v$,
 \item the critical value $-v$ escapes to $\infty$, or
 \item $-v$ has bounded orbit and there exists an integer $N \ge 1$, such that for every periodic Fatou component $U$, the closure of every strictly preperiodic component of $f^{-N}(U)$ is disjoint from the $\omega$-limit set of $-v$.
 \end{enumerate}
 Then $J(f) \cup \{\infty\}$ is locally connected.
\end{theorem}

The positions of critical points and critical values satisfy the following dichotomy:
\begin{lemma}\label{lm: dichotomy of the positions of critical points}
 Let $f$ be a cosine function. Suppose that $F(f)$ is non-empty, $f$ has no Siegel disks, and the critical orbits either escape to $\infty$ or remain bounded. Then one of the following holds.
 \begin{enumerate}
 \item The critical points of $f$ lie in the unique Fatou component, and all Fatou components are unbounded;
 \item Every component of $F(f)$ contains at most one critical point of $f$, and every component of $F(f)$ is bounded.
 \end{enumerate}
\end{lemma}
\begin{proof}
 We will discuss the positions of critical values of $f$. Let $B$ be a Fatou component of $f$. Since $F(f)$ has no wandering components, we assume that $B$ is periodic. Moreover, we consider an invariant Fatou component $B$, i.e. $f(B)=B$.

\begin{proposition}\label{prop: bounded periodic Fatou components are Jordan domains}
 Suppose that $P(f)$ is bounded, $U$ is a bounded periodic Fatou component of $f$, which is not a Siegel disk. Then $U$ is a Jordan domain.
\end{proposition}
\begin{proof}
 Let $B=f^q(U)$ be the Fatou component in the cycle of $U$, which contains a critical point. Let $z \in \partial B \setminus \Gamma_{\infty}$ so that $\hat{P}_n(z)$, $n \ge 1$, are well-defined. Consider $T(z)$. Let
 \begin{equation*}
 d_l(z)=\min\{n \ge 1: \hat{P}_n(f^l(z)) \text{ contains a critical point}\}.
 \end{equation*}

\begin{proof}[Proof of Theorem \ref{thm: bounded Fatou components are Jordan domains}]
 Let $U$ be a bounded Fatou component of $f$, which is not eventually mapped to a Siegel disk. By \cite{EL92}, Theorem 1 \& Theorem 3, there exists minimal $n \ge 0$ such that $f^n(U)$ is a periodic Fatou component of $f$, and $f^n(U)$ is bounded. By Proposition \ref{prop: bounded periodic Fatou components are Jordan domains}, $f^n(U)$ is a Jordan domain. Since $f^n: \partial U \to \partial f^n(U)$ is a local homeomorphism, $\partial U$ is also locally connected, and $U$ is a Jordan domain by the Maximum Modulus Principle.
\end{proof}

\begin{corollary}\label{cor: Fatou components are locally connected in renormalizable case}
    Let $f$ be a cosine function. Suppose that the Fatou set is non-empty, and one of the critical values escapes to $\infty$. If $f$ has no Siegel disks, then all Fatou components of $f$ are Jordan domains.
\end{corollary}

\begin{theorem}\label{thm: bounded Fatou components are Jordan domains}
    Let $f(z)=ae^z+be^{-z}$ be a cosine function. Suppose that $P(f)$ is bounded, $U$ is a bounded Fatou component, which is not eventually mapped to a Siegel disk. Then $U$ is a Jordan domain.
\end{theorem}

It is worth noting that there are also some results about the local connectivity of the boundaries of Siegel disks. For example, for quadratic polynomials, if the rotation number of the Siegel disk is of bounded type, then the boundary of the disk is locally connected (\cite{Pet96}). And the local connectivity of Siegel disks holds for some families of transcendental entire functions (\cite{Zhang05,Yang13,Zhang16,ZFS20}).

The cosine function $f(z)=ae^{z}+be^{-z}$ has two critical values and no asymptotic values. So it shares some similarities with cubic polynomials. It is known that for a polynomial $p$ with at least one escaping critical orbit, or equivalently with $J(p)$ disconnected, $J(p)$ is a Cantor set if and only if the critical components of $K(p)$ is not periodic (\cite{BH92,QY09}). Here, $K(p)$, which is called the filled-in Julia set, is the complement of the immediate basin of $\infty$. In other words, if $p$ has periodic Fatou components and $J(p)$ is disconnected, then $p$ is renormalizable. We have the analogy for cosine functions:
\begin{theorem}\label{thm: cosine functions with an escaping critical value is renormalizable}
    Let $f$ be a cosine function. Suppose that the Fatou set is non-empty, and one of the critical values escapes to $\infty$. Then $f$ is renormalizable.
\end{theorem}

In this case, the local connectivity of Fatou components can be deduced easily.
\begin{corollary}\label{cor: Fatou components are locally connected in renormalizable case}
    Let $f$ be a cosine function. Suppose that the Fatou set is non-empty, and one of the critical values escapes to $\infty$. If $f$ has no Siegel disks, then all Fatou components of $f$ are Jordan domains.
\end{corollary}

Finally, with Theorem \ref{thm: bounded Fatou components are Jordan domains} and Corollary \ref{cor: Fatou components are locally connected in renormalizable case}, we conclude the local connectivity of $J(f)$:
\begin{theorem}\label{thm: local connectivity of Julia set}
    Let $f$ be a cosine function. Suppose that $F(f)$ is non-empty and has no Siegel disks. Suppose that a critical value $v$ lies in $F(f)$. Suppose that $f$ satisfies one of the following conditions:
    \begin{enumerate}
        \item the critical value $-v \in F(f)$ and is not in the same Fatou component as $v$,
        \item the critical value $-v$ escapes to $\infty$, or
        \item $-v$ has bounded orbit and there exists an integer $N \ge 1$, such that for every periodic Fatou component $U$, the closure of every strictly preperiodic component of $f^{-N}(U)$ is disjoint from the $\omega$-limit set of $-v$.
    \end{enumerate}
    Then $J(f) \cup \{\infty\}$ is locally connected.
\end{theorem}

This paper is structured as following:
Section \ref{sec: basic settings} stated some basic settings and proved a dichotomy of the boundedness of Fatou components; Section \ref{sec: construction of puzzles} constructed the Yoccoz puzzle for cosine functions with bounded post-critical set; Section \ref{sec: local connectivity of bounded Fatou components} proved Theorem \ref{thm: bounded Fatou components are Jordan domains} using the puzzle; Section \ref{sec: renormalization of cosine functions} proved Theorem \ref{thm: cosine functions with an escaping critical value is renormalizable} and Corollary \ref{cor: Fatou components are locally connected in renormalizable case}; and Section \ref{sec: local connectivity of Julia sets} proved Theorem \ref{thm: local connectivity of Julia set}.

The post-theorem introduction indicates that Theorem~\ref{thm: bounded Fatou components are Jordan domains} is proved "using the puzzle": specifically, it says that Section~\ref{sec: construction of puzzles} "constructed the Yoccoz puzzle for cosine functions with bounded post-critical set" and Section~\ref{sec: local connectivity of bounded Fatou components} "proved Theorem~\ref{thm: bounded Fatou components are Jordan domains} using the puzzle."

\label{thm: bounded Fatou components are Jordan domains}
    Let $f(z)=ae^z+be^{-z}$ be a cosine function. Suppose that $P(f)$ is bounded, $U$ is a bounded Fatou component, which is not eventually mapped to a Siegel disk. Then $U$ is a Jordan domain.,

Question: Let \(f(z)=ae^z+be^{-z}\) with \(a,b\in\mathbb C\setminus\{0\}\), so \(f\) is a cosine function. Let \(P(f)\) denote the postsingular set of \(f\), i.e. the closure of the forward orbits of its singular values; assume that \(P(f)\) is bounded in \(\mathbb C\). Let \(U\) be a bounded Fatou component of \(f\) (a connected component of the Fatou set), and assume that \(U\) is not eventually mapped to a Siegel disk, meaning that there is no \(n\ge 0\) such that \(f^n(U)\) is a Siegel disk. Which conclusion about \(U\) holds under these assumptions?

\(U\) is a Jordan domain; equivalently, its boundary \(\partial U\) is a Jordan curve.

Choices:
B. \(U\) is simply connected, and \(\partial U\) is locally connected, but \(U\) need not be a Jordan domain.
C. \(\partial U\) is locally connected.
D. Every bounded Fatou component of \(f\) is a Jordan domain, even without assuming that \(P(f)\) is bounded or that \(U\) is not eventually mapped to a Siegel disk.
E. If some iterate \(f^n(U)\) is a bounded periodic Fatou component that is not a Siegel disk, then that periodic component is a Jordan domain, but one cannot in general conclude that \(U\) itself is a Jordan domain.

upgraded conclusion from local connectivity to Jordan-domain equivalence is removed

dropped the Jordan-domain conclusion and kept only local connectivity of the boundary

boundedness of postsingular set and exclusion of eventual Siegel disks

pullback/local-homeomorphism step from periodic image back to the original component


[Rigidity of Siegel polynomials of bounded type]
\label{Thm:A}
Let $f \colon \mathbb C \to \mathbb C$ be an atomic Siegel polynomial of bounded type and degree $d \ge 2$. Then:

\begin{enumerate}[\rm (A)]
\item
\label{it:main:1}
each point in the Julia set $J_f$ is dynamically rigid (i.e., the orbits of any two orbits can be distinguished in terms of symbolic dynamics); in particular, $J_f$ is locally connected.
\item
\label{it:main:2}
$J_f$ is quasiconformally rigid (i.e., it does not admit non-trivial invariant line fields).
\item
\label{it:main:3}
If $\widetilde f$ is another atomic Siegel polynomial of bounded-type that is combinatorially equivalent to $f$, then $f$ and $\widetilde f$ are affinely conjugate.
\end{enumerate}

\label{sec:intro}

A central theme in one-dimensional complex dynamics is the study of \emph{rigidity phenomena}, that is, understanding to what extent the combinatorial or topological data associated with a dynamical system determines its conformal or analytic structure. This perspective has driven substantial progress in the field, tracing back to the seminal work of Douady and Hubbard \cite{DoHu}, who introduced the notion of polynomial-like mappings and used them to establish deep structural results about the Mandelbrot set and quadratic polynomials. The landmark results of Yoccoz \cite{HY} later established rigidity for a broad class of quadratic polynomials, showing that combinatorial data (the landing pattern of external rays and normalization in the Fatou set) suffice to determine the conformal conjugacy class of the map. These ideas were subsequently extended to higher-degree polynomials, most notably through the development of enhanced puzzle techniques and renormalization theory in the works of Lyubich \cite{Ly}, Lyubich and Kahn \cite{KaLy1, KaLy2}, Avila, Kahn, Lyubich, and Shen \cite{AKLS}, Kozlovski, Shen, van Strien \cite{KSS}, Kozlovski and van Strien \cite{KvS}, and others.

However, rigidity results often rely on the absence of neutral behaviour in the dynamics. In cases where the dynamics has neutral periodic points, the situation becomes more delicate. For instance, in the parabolic setting, powerful tools like parabolic implosion and Lavaurs maps allow for a fine control of the local and global dynamics, and rigidity results can often be recovered. In contrast, in the presence of irrationally indifferent periodic points, the problem becomes substantially more subtle. One of the better-understood scenarios within neutral dynamics arises in the case of Siegel disks with bounded type rotation numbers.

Let $f \colon \mathbb C \to \mathbb C$ be a polynomial with a fixed point at $0$. We say that $0$ is \emph{irrationally indifferent} (with {\it rotation number} $\theta  \in (\mathbb R \setminus \mathbb Q) / \mathbb Z$) if $f'(0) = e^{2\pi i \theta}$. If additionally the dynamics of $f$ is linearizable locally near $0$, then $0$ is called a {\it Siegel point}. This means that in some neighbourhood of $0$ the map $f$ is conformally conjugate to the rigid rotation $z \mapsto e^{2\pi i \theta} \cdot z$. In this case, the maximal domain of definition for such a conjugacy is called the \emph{Siegel disk}, denoted $\Delta_f \ni 0$, and $f$ is referred to as a {\it Siegel polynomial}. Lastly, the rotation number, the Siegel point, the Siegel disk and the Siegel polynomial are said to be \emph{of bounded type} if all entries in the continued fraction expansion of $\theta$ are uniformly bounded.

The first seminal result on Siegel polynomials was due to Douady--Ghys--Herman--Shishikura (see \cite{Do, He2}). Using quasi-surgery, they constructed a Blaschke product model for the dynamics of $f$ in case $f$ is quadratic ($\deg(f) = 2$) and the rotation number is of bounded type. This allowed them to conclude that the Siegel boundary $\partial \Delta_f$ is a quasi-circle containing at least one critical point (which is unique in the quadratic case). This result was generalized to the cubic case $\deg(f) =3$ by Zakeri in \cite{Za}, then to any degree by Shishikura and Zhang \cite{Zh} (see Theorem~\ref{siegel bound quasi}). (See also \cite{WaYaZhZh}.)

The rigidity of Siegel polynomials of bounded type was settled in the quadratic case by Petersen in \cite{Pe} (see also the work of Yampolsky \cite{Ya1}). Petersen proved that the Julia set $J_f$ of a quadratic Siegel polynomial $f$ of bounded type is locally connected, thereby showing that the topology of the dynamics of $f$ is entirely determined by its combinatorial data. He then showed that $J_f$ has zero measure, which implies that it has a trivial conformal conjugacy class.

In Petersen's work, it is crucial that $\deg(f) = 2$, so that the unique critical point is forced to be contained in the Siegel boundary $\partial \Delta_f$. In the case $\deg(f) \geq 3$, it is possible that there exists an external, recurrent critical point whose orbit accumulates to $\partial \Delta_f$, but does not land exactly on it. In this highly non-linear situation, the geometry of $\partial \Delta_f$ could, a priori, become highly distorted, causing a failure of rigidity. This problem is addressed in \cite{Y}, where it is shown that even in the higher degree case, local connectivity still holds at $\partial \Delta_f$. This work is followed in \cite{YZh}, where it is shown that global rigidity holds for cubic Siegel polynomials of bounded type.

In this paper, we address the rigidity of Siegel polynomials of bounded type for arbitrary degrees. We focus on polynomials that are non-renormalizable in the sense of Douady and Hubbard. This means that the dynamics cannot be decomposed into simpler, more basic parts that can be analyzed separately. We say that such maps are \emph{atomic}.

The main result of this paper is the following rigidity theorem for atomic Siegel polynomials of bounded type. This is the first time in the literature that the neutral irrational dynamics is incorporated into the rigidity picture for polynomial maps of arbitrary degree.

Let $f \colon \mathbb C \to \mathbb C$ be a polynomial with a fixed point at $0$. We say that $0$ is \emph{irrationally indifferent} (with {\it rotation number} $\theta \in (\mathbb R \setminus \mathbb Q) / \mathbb Z$) if $f'(0) = e^{2\pi i \theta}$. If additionally the dynamics of $f$ is linearizable locally near $0$, then $0$ is called a {\it Siegel point}. This means that in some neighbourhood of $0$ the map $f$ is conformally conjugate to the rigid rotation $z \mapsto e^{2\pi i \theta} \cdot z$. In this case, the maximal domain of definition for such a conjugacy is called the \emph{Siegel disk}, denoted $\Delta_f \ni 0$, and $f$ is referred to as a {\it Siegel polynomial}. Lastly, the rotation number, the Siegel point, the Siegel disk and the Siegel polynomial are said to be \emph{of bounded type} if all entries in the continued fraction expansion of $\theta$ are uniformly bounded.

The first seminal result on Siegel polynomials was due to Douady--Ghys--Herman--Shishikura (see \cite{Do, He2}). Using quasi-surgery, they constructed a Blaschke product model for the dynamics of $f$ in case $f$ is quadratic ($\deg(f) = 2$) and the rotation number is of bounded type. This allowed them to conclude that the Siegel boundary $\partial \Delta_f$ is a quasi-circle containing at least one critical point (which is unique in the quadratic case). This result was generalized to the cubic case $\deg(f) =3$ by Zakeri in \cite{Za}, then to any degree by Shishikura and Zhang \cite{Zh} (see Theorem~\ref{siegel bound quasi}). (See also \cite{WaYaZhZh}.)

The rigidity of Siegel polynomials of bounded type was settled in the quadratic case by Petersen in \cite{Pe} (see also the work of Yampolsky \cite{Ya1}). Petersen proved that the Julia set $J_f$ of a quadratic Siegel polynomial $f$ of bounded type is locally connected, thereby showing that the topology of the dynamics of $f$ is entirely determined by its combinatorial data. He then showed that $J_f$ has zero measure, which implies that it has a trivial conformal conjugacy class.

In Petersen's work, it is crucial that $\deg(f) = 2$, so that the unique critical point is forced to be contained in the Siegel boundary $\partial \Delta_f$. In the case $\deg(f) \geq 3$, it is possible that there exists an external, recurrent critical point whose orbit accumulates to $\partial \Delta_f$, but does not land exactly on it. In this highly non-linear situation, the geometry of $\partial \Delta_f$ could, a priori, become highly distorted, causing a failure of rigidity. This problem is addressed in \cite{Y}, where it is shown that even in the higher degree case, local connectivity still holds at $\partial \Delta_f$. This work is followed in \cite{YZh}, where it is shown that global rigidity holds for cubic Siegel polynomials of bounded type.

In this paper, we address the rigidity of Siegel polynomials of bounded type for arbitrary degrees. We focus on polynomials that are non-renormalizable in the sense of Douady and Hubbard. This means that the dynamics cannot be decomposed into simpler, more basic parts that can be analyzed separately. We say that such maps are \emph{atomic}.

The main result of this paper is the following rigidity theorem for atomic Siegel polynomials of bounded type. This is the first time in the literature that the neutral irrational dynamics is incorporated into the rigidity picture for polynomial maps of arbitrary degree.

\begin{thm}[Shishikura, Zhang]\label{siegel bound quasi}
The Siegel boundary $\partial \Delta_f$ is a quasi-circle that contains at least one critical point $c_0$ of $f$. \qed
\end{thm}

\maketitle
\begin{abstract}
In this paper, we study rigidity of polynomials of arbitrary degree in the presence of neutral dynamics. Specifically, we focus on \emph{non-renormalizable} (in the sense of Douady and Hubbard) complex polynomials of degree $d \ge 2$ that possess a Siegel disk of bounded type rotation number. We refer to such maps as \emph{atomic Siegel polynomials of bounded type}. In this setting, our main results are:
\smallskip
\begin{enumerate}
\item[(A)] Atomic Siegel polynomials of bounded type have locally connected Julia sets;
\item[(B)] These Julia sets are quasiconformally rigid, i.e., they do not support invariant line fields;
\item[(C)] Any two combinatorially equivalent atomic Siegel polynomials of bounded type coincide up to an affine change of coordinates.
\end{enumerate}
\smallskip
In particular, item (C) verifies the notorious \emph{Combinatorial Rigidity Conjecture} for atomic Siegel polynomials of bounded type and arbitrary degree. By bringing neutral Siegel dynamics into the picture, we extend the celebrated higher-degree rigidity theorems of Avila--Kahn--Lyubich--Shen and Kozlovski--van Strien, which until now applied only in the Yoccoz setting, i.e., for finitely many times renormalizable polynomials without irrationally indifferent periodic points.
\end{abstract}

The main result of this paper is the following rigidity theorem for atomic Siegel polynomials of bounded type. This is the first time in the literature that the neutral irrational dynamics is incorporated into the rigidity picture for polynomial maps of arbitrary degree.

We will define all the concepts in the follow-up sections; in particular, combinatorial equivalence is defined in Section~\ref{sec:topmodel} (see Definition~\ref{Def:CombEq}). This result extends a recent result due to the second author and Zhang \cite{YZh} on rigidity of bounded-type Siegel polynomials of degree $3$ (one free critical point). Main Theorem~\eqref{it:main:3} confirms the \emph{Combinatorial Rigidity Conjecture} in the above-mentioned class of polynomials. This conjecture asserts that for polynomials under an \emph{appropriate} definition of the combinatorial equivalence, combinatorial rigidity holds. We remark that our definition includes the condition that $f$ and $\tilde f$ are \emph{Fatou normalized} and the positions of critical points on the boundary of the Siegel disk are combinatorially the same, see Section~\ref{sec:fatoutrees} for details. Since we assume that $f, \tilde f$ are non-renormalizable, their rational laminations, i.e., the combinatorial pattern of landing external rays, are empty. In our context, Fatou normalization means that each critical point $c$ in a preimage of a Siegel disk $\Delta_f$ lands at $0$.

\begin{thm}[Parameter rigidity]
\label{Thm:BoxParamRigidity}
Let $F\colon \U\to \V$ be a non-renormalizable dynamically natural complex box mapping. Suppose $\tilde F \colon \tilde \U \to \tilde \V$ is another dynamically natural complex box mapping for which there exists a quasiconformal homeomorphism $H \colon \V \to \tilde\V$ so that
\begin{enumerate}
\item
\label{part3c}
$\tilde F$ is combinatorially equivalent to $F$ w.r.t.\ $H$, and so in particular $H(\U) = \tilde \U$,
\item
\label{part3b}
$\tilde F \circ H = H \circ F$ on $\partial \U$, i.e.\ $H$ is a conjugacy on $\partial \U$.
\end{enumerate}
Then $F$ and $\tilde F$ are quasiconformally conjugate, and this conjugacy agrees with $H$ on $\V\setminus \U$. \qed
\end{thm}

\begin{thm}
\label{Thm:DynRigidity}
Let $f$ be an atomic Siegel polynomial of bounded type. Then
\begin{enumerate}
\item
all fibers of points in the Julia set $J(f)$ are trivial, and hence, $J(f)$ is locally connected;
\item
$J(f)$ carries no invariant line fields.
\end{enumerate}
\end{thm}

\begin{defn}[Combinatorial equivalence]
\label{Def:CombEq}
Two atomic Siegel polynomials $f, \tif : \bbC \to \bbC$ of bounded type are said to be {\it combinatorially equivalent} if
\begin{enumerate}
 \item
 $\deg(f) = \deg(\tif\,)$, $\rho(f)=\rho(\tif\,)$;
 \item
 $\adr_{\crit}(f)=\adr_{\crit}(\tif)$ (see Definition~\ref{Def:CritAdr});
 \item
 the orbits of each pair $c \in \Crit(f) \cap \inter(K_f)$, $\tilde c\in \Crit(\tif\,) \cap \inter(K_{\tif})$ of the corresponding critical points land in $\Delta_f$, $\Delta_{\tif}$ in the same conformal position with respect to the uniformizing coordinates.
\end{enumerate}
\end{defn}

\begin{thm}[Topological rigidity]\label{poly topo rigid}
Let $f, \tif : \bbC \to \bbC$ be atomic Siegel polynomials of bounded type. If $f$ and $\tif$ are combinatorially equivalent, then they are topologically conjugate, and, restricted to the Fatou sets, this conjugacy is conformal.\qed
\end{thm}

\begin{prop}
\label{Prop:CombEquivalent}
If $f$ and $\tf$ are two topologically conjugate, atomic Siegel polynomials of bounded type, then in Theorem~\ref{Thm:Extract} one can choose dynamically natural complex box mappings $F \colon \U \to \V$ and $\tilde F \colon \tilde \U \to \tilde \V$ to be combinatorially equivalent in the sense of Definition~\ref{DefA:CombEquivBoxMappings}, with respect to some quasiconformal homeomorphism $H \colon \V \to \tilde \V$ such that $H(\U) = \tilde \U$ and $\tilde F \circ H = H \circ F$ on $\partial \U$. \qed
\end{prop}

\begin{defn}[Combinatorial equivalence]
\label{Def:CombEq}
Two atomic Siegel polynomials $f, \tif : \bbC \to \bbC$ of bounded type are said to be {\it combinatorially equivalent} if
\begin{enumerate}
    \item
    $\deg(f) = \deg(\tif\,)$, $\rho(f)=\rho(\tif\,)$;
    \item
    $\adr_{\crit}(f)=\adr_{\crit}(\tif)$ (see Definition~\ref{Def:CritAdr});
    \item
    the orbits of each pair $c \in \Crit(f) \cap \inter(K_f)$, $\tilde c\in \Crit(\tif\,) \cap \inter(K_{\tif})$ of the corresponding critical points land in $\Delta_f$, $\Delta_{\tif}$ in the same conformal position with respect to the uniformizing coordinates.
\end{enumerate}
\end{defn}

\begin{enumerate}[\rm (A)]
\item
\label{it:main:1}
each point in the Julia set $J_f$ is dynamically rigid (i.e., the orbits of any two orbits can be distinguished in terms of symbolic dynamics); in particular, $J_f$ is locally connected.
\item
\label{it:main:2}
$J_f$ is quasiconformally rigid (i.e., it does not admit non-trivial invariant line fields).
\item
\label{it:main:3}
If $\tilde f$ is another atomic Siegel polynomial of bounded-type that is combinatorially equivalent to $f$, then $f$ and $\tilde f$ are affinely conjugate.
\end{enumerate}

We will define all the concepts in the follow-up sections; in particular, combinatorial equivalence is defined in Section~\ref{sec:topmodel} (see Definition~\ref{Def:CombEq}). This result extends a recent result due to the second author and Zhang \cite{YZh} on rigidity of bounded-type Siegel polynomials of degree $3$ (one free critical point). Main Theorem~\eqref{it:main:3} confirms the \emph{Combinatorial Rigidity Conjecture} in the above-mentioned class of polynomials. This conjecture asserts that for polynomials under an \emph{appropriate} definition of the combinatorial equivalence, combinatorial rigidity holds. We remark that our definition includes the condition that $f$ and $\widetilde f$ are \emph{Fatou normalized} and the positions of critical points on the boundary of the Siegel disk are combinatorially the same, see Section~\ref{sec:fatoutrees} for details. Since we assume that $f, \widetilde f$ are non-renormalizable, their rational laminations, i.e., the combinatorial pattern of landing external rays, are empty. In our context, Fatou normalization means that each critical point $c$ in a preimage of a Siegel disk $\Delta_f$ lands at $0$.

\begin{rem}
Our Main Theorem can be easily extended to the cases where one allows for finitely many orbits of periodic Siegel disks and finitely many renormalizations. In those cases, in the definition of combinatorial equivalence one needs to include in the combinatorial data also the landing patterns of external and internal rays. Furthermore, one can also include cycles of attracting and parabolic periodic points.

For clarity of theexposition, in this paper, we restrict our attention to atomic Siegel polynomials.
\end{rem}

\subsection{Outline of the proof}

The proof uses the following strategy, which was successful in a number of cases (see \cite[Section 2]{CDKvS}), including the one in the current paper:

\begin{enumerate}
\item
\emph{Markov partition}: In a given family $\mathcal F$ of mappings and a map $f \in \mathcal F$ in this family, construct a puzzle partition (i.e., a Yoccoz- or Markov-type partition) around points in the Julia set $J_f$ of $f$. In the context of the current paper, the partition is given in terms of Siegel bubble rays extended by external rays, see Section~\ref{sec:puzzle}.

\item
\emph{Rigidity on the Julia boundary of puzzle pieces}: Establish (dynamical) rigidity on the boundary of each puzzle piece, i.e., for a puzzle piece $P$, show that each point in $\partial P \cap J_f$ can be surrounded by a union of arbitrary small puzzle pieces. In our case, this result was proven in \cite{Y} (see Section~\ref{sec:puzzle}).

\item
\emph{Extract a dynamically natural complex box mapping}: Construct a dynamically natural box mapping (in the sense of Section~\ref{sec:box}) around the critical points of $f$ that do not land on the puzzle boundary. This mapping controls the dynamics of most critical points, except for those whose orbits accumulate only at the puzzle boundary. In our paper, the construction of such a mapping is carried out in Section~\ref{sec:DNBM}.

\item
\emph{Local connectivity and quasiconformal rigidity of the Julia set}: Use the rigidity result for box mappings (i.e., puzzle pieces shrink to points, no invariant line fields) to infer the corresponding results for $f \in \mathcal F$. In our case, this is done in Section~\ref{sec:proofAB}, where we prove the items \eqref{it:main:1} and \eqref{it:main:2} of the Main Theorem.

\item
\emph{Topological model of the Julia set}: Construct topological models of the dynamics for maps $f \in \mathcal F$ and prove that they are determined uniquely by combinatorial data. This is done in Section~\ref{sec:combtrees} and \ref{sec:fatoutrees}.

\item
\emph{Topological rigidity of combinatorially equivalent maps}: Prove that the filled Julia sets of maps in $\mathcal F$ are homeomorphic to the topological models constructed in the previous step. It follows that two maps within the same combinatorial class must be topologically conjugate. This is carried out in Section~\ref{sec:topmodel}.

\item
\emph{Conformal rigidity of topologically equivalent maps}: Use quasiconformal rigidity of the corresponding complex box mappings for the topologically conjugate $f, \widetilde f$ to conclude that $f, \widetilde f$ are quasiconformally conjugate, and hence, because of quasiconformal rigidity of $J_f, J_{\widetilde f}$ and Fatou normalization, $f, \widetilde f$ are conformally (affine) conjugate.

In our paper, this step is done in Section~\ref{sec:top2qc}. This concludes the proof of Main Theorem~\eqref{it:main:3}.
\end{enumerate}

The post-theorem introduction contains an explicit \subsection{Outline of the proof} giving the strategy (successful in other cases, cf. \cite[Section 2]{CDKvS}):
\begin{enumerate}
\item \emph{Markov partition}: for $f$ in the family $\mathcal F$, construct a puzzle (Yoccoz-/Markov-type) partition around points in $J_f$; here it is “given in terms of Siegel bubble rays extended by external rays” (Section~\ref{sec:puzzle}).
\item \emph{Rigidity on the Julia boundary of puzzle pieces}: prove dynamical rigidity on $\partial P$ for each puzzle piece $P$, i.e. each point of $\partial P\cap J_f$ can be surrounded by “a union of arbitrary small puzzle pieces”; in this setting it is cited as proven in \cite{Y} (Section~\ref{sec:puzzle}).
\item \emph{Extract a dynamically natural complex box mapping}: build a “dynamically natural box mapping” (Section~\ref{sec:box}) around critical points not landing on the puzzle boundary; it “controls the dynamics of most critical points, except for those whose orbits accumulate only at the puzzle boundary” (Section~\ref{sec:DNBM}).
\item \emph{Local connectivity and quasiconformal rigidity of the Julia set}: apply rigidity for box mappings (“puzzle pieces shrink to points, no invariant line fields”) to obtain the corresponding statements for $f$; this yields items \eqref{it:main:1} and \eqref{it:main:2} in Section~\ref{sec:proofAB}.
\item \emph{Topological model of the Julia set}: construct topological models for $f\in\mathcal F$ and show they are “determined uniquely by combinatorial data” (Sections~\ref{sec:combtrees} and \ref{sec:fatoutrees}).
\item \emph{Topological rigidity of combinatorially equivalent maps}: show filled Julia sets are homeomorphic to these models, so “two maps within the same combinatorial class must be topologically conjugate” (Section~\ref{sec:topmodel}).
\item \emph{Conformal rigidity of topologically equivalent maps}: use quasiconformal rigidity of the associated box mappings for topologically conjugate $f,\widetilde f$ to get a quasiconformal conjugacy; then, using “quasiconformal rigidity of $J_f, J_{\widetilde f}$ and Fatou normalization,” conclude $f,\widetilde f$ are “conformally (affine) conjugate” (Section~\ref{sec:top2qc}), which “concludes the proof of Main Theorem~\eqref{it:main:3}.”
\end{enumerate}

The post-theorem introduction contains an explicit \subsection{Outline of the proof} giving the strategy (successful in other cases, cf. \cite[Section 2]{CDKvS}):
\begin{enumerate}
\item \emph{Markov partition}: for $f$ in the family $\mathcal F$, construct a puzzle (Yoccoz-/Markov-type) partition around points in $J_f$; here it is “given in terms of Siegel bubble rays extended by external rays” (Section~\ref{sec:puzzle}).
\item \emph{Rigidity on the Julia boundary of puzzle pieces}: prove dynamical rigidity on $\partial P$ for each puzzle piece $P$, i.e. each point of $\partial P\cap J_f$ can be surrounded by “a union of arbitrary small puzzle pieces”; in this setting it is cited as proven in \cite{Y} (Section~\ref{sec:puzzle}).
\item \emph{Extract a dynamically natural complex box mapping}: build a “dynamically natural box mapping” (Section~\ref{sec:box}) around critical points not landing on the puzzle boundary; it “controls the dynamics of most critical points, except for those whose orbits accumulate only at the puzzle boundary” (Section~\ref{sec:DNBM}).
\item \emph{Local connectivity and quasiconformal rigidity of the Julia set}: apply rigidity for box mappings (“puzzle pieces shrink to points, no invariant line fields”) to obtain the corresponding statements for $f$; this yields the following two conclusions:
\begin{enumerate}[\rm (A)]
\item
\label{it:main:1}
each point in the Julia set $J_f$ is dynamically rigid (i.e., the orbits of any two orbits can be distinguished in terms of symbolic dynamics); in particular, $J_f$ is locally connected.
\item
\label{it:main:2}
$J_f$ is quasiconformally rigid (i.e., it does not admit non-trivial invariant line fields).
\item
\label{it:main:3}
If $\tilde f$ is another atomic Siegel polynomial of bounded-type that is combinatorially equivalent to $f$, then $f$ and $\tilde f$ are affinely conjugate.
\end{enumerate}
 in Section~\ref{sec:proofAB}.
\item \emph{Topological model of the Julia set}: construct topological models for $f\in\mathcal F$ and show they are “determined uniquely by combinatorial data” (Sections~\ref{sec:combtrees} and \ref{sec:fatoutrees}).
\item \emph{Topological rigidity of combinatorially equivalent maps}: show filled Julia sets are homeomorphic to these models, so “two maps within the same combinatorial class must be topologically conjugate” (Section~\ref{sec:topmodel}).
\item \emph{Conformal rigidity of topologically equivalent maps}: use quasiconformal rigidity of the associated box mappings for topologically conjugate $f,\widetilde f$ to get a quasiconformal conjugacy; then, using “quasiconformal rigidity of $J_f, J_{\widetilde f}$ and Fatou normalization,” conclude $f,\widetilde f$ are “conformally (affine) conjugate” (Section~\ref{sec:top2qc}), which “concludes the proof of Main Theorem~\eqref{it:main:3}.”
\end{enumerate}

Question: Let \(f:\mathbb C\to\mathbb C\) be an atomic Siegel polynomial of bounded type and degree \(d\ge 2\), meaning that \(f\) is a non-renormalizable complex polynomial of degree at least \(2\) with a Siegel disk whose irrational rotation number is of bounded type (equivalently, the continued-fraction coefficients of the rotation number are uniformly bounded). For another atomic Siegel polynomial of bounded type \(\widetilde f\), say that \(f\) and \(\widetilde f\) are combinatorially equivalent if they have the same degree and rotation number, the same critical-address data, and the orbits of corresponding critical points in the interiors of their filled Julia sets land in their Siegel disks in the same conformal position with respect to uniformizing coordinates. Under these assumptions, which statement about \(J_f\) and about such a map \(\widetilde f\) holds?

Each point of the Julia set \(J_f\) is dynamically rigid (that is, the orbits of any two orbits can be distinguished in terms of symbolic dynamics), and in particular \(J_f\) is locally connected; moreover, \(J_f\) is quasiconformally rigid, meaning that it admits no non-trivial invariant line fields; and if \(\widetilde f\) is another atomic Siegel polynomial of bounded type that is combinatorially equivalent to \(f\), then \(f\) and \(\widetilde f\) are affinely conjugate.

Choices:
B. Each point of the Julia set \(J_f\) is dynamically rigid, and in particular \(J_f\) is locally connected; moreover, \(J_f\) is quasiconformally rigid; and if \(\widetilde f\) is another atomic Siegel polynomial of bounded type with the same degree and rotation number as \(f\), then \(f\) and \(\widetilde f\) are affinely conjugate.
C. The Julia set \(J_f\) is quasiconformally rigid, meaning that it admits no non-trivial invariant line fields; and if \(\widetilde f\) is another atomic Siegel polynomial of bounded type that is combinatorially equivalent to \(f\), then \(f\) and \(\widetilde f\) are topologically conjugate.
D. Each point of the Julia set \(J_f\) is dynamically rigid, and in particular \(J_f\) is locally connected; moreover, \(J_f\) is quasiconformally rigid, meaning that it admits no non-trivial invariant line fields; and if \(\widetilde f\) is another atomic Siegel polynomial of bounded type that is combinatorially equivalent to \(f\), then there exists a quasiconformal conjugacy between \(f\) and \(\widetilde f\).
E. Each point of the Julia set \(J_f\) that does not lie on the boundary of a puzzle piece is dynamically rigid, and in particular \(J_f\) is locally connected; moreover, \(J_f\) is quasiconformally rigid away from the Siegel boundary; and if \(\widetilde f\) is another atomic Siegel polynomial of bounded type that is combinatorially equivalent to \(f\), then \(f\) and \(\widetilde f\) are affinely conjugate whenever all critical orbits avoid accumulation on the puzzle boundary.

full combinatorial-equivalence data required for rigidity

affinely conjugate weakened to topologically conjugate; dynamic rigidity/local connectivity clause dropped

final affine conclusion weakened to quasiconformal-conjugacy endpoint

boundary-vs-nonboundary critical orbit split in box-mapping extraction