On recurrence and entropy in hyperspace of continua in dimension one

TL;DR

The paper addresses whether entropy is preserved when passing from a graph map $f$ to its induced map on the continua hyperspace, proving $h_{top}(f)=h_{top}( ilde{f})$ for topological graphs. The approach hinges on a detailed analysis of Rec$( ilde{f})$ via ω-limit structures, showing that most recurrent continua are periodic unless tied to circumferential or solenoidal behavior, which are then analyzed through almost-conjugate models to irrational rotations. This allows localization of entropy sources and a variational-entropy argument that rules out entropy inflation on $C(G)$. The result extends known interval/tree cases to graphs and clarifies how one-dimensional dynamics constrain entropy growth on continua, providing a framework for entropy computation via ω-limit typology in one-dimensional settings.

Abstract

We show that if $G$ is a topological graph, and $f$ is continuous map, then the induced map $\tilde{f}$ acting on the hyperspace $C(G)$ of all connected subsets of $G$ by natural formula $\tilde{f}(C)=f(C)$ carries the same entropy as $f$. This is well known that it does not hold on the larger hyperspace of all compact subsets. Also negative examples were given for the hyperspace $C(X)$ on some continua $X$, including dendrites. Our work extends previous positive results obtained first for much simpler case of compact interval by completely different tools.

Figures (3)

• Figure 1: Sketch for the case $y\notin E_0$: $K^{n_{k+r}}_{i^{k+r}_y}$ is not necessarily an arc or a subset of $E_0$. $L$ is subset of $K^{n_{k-1}}_{i^{k-1}_y}$ which is invariant under $f^{p_{n_k}}$ and does not contain $E_0$ so $L$ has to grow. Also, the order of $K_{i_A^{k+r}}^{n_{k+r}}$, $K_{i_A^{k}}^{n_{k}}$ and $K_{i_y^{k+r}}^{n_{k+r}}\cap C$ in $E$ is shown.
• Figure 2: The image of a factor map is shown. $E$ is the edge with endpoints $b_1,b_2.$ All the points of $\omega\cap E$ lie between $x_1$ and $x_2$ in the order of $E$. Figure \ref{['fig:first']} illustrates the case when $M\subset E$ while Figure \ref{['fig:second']} illustrates the case when $[x_1,x_2]\setminus M\neq \emptyset$ so $\{b_1,b_2\}\subset M,$ hence $\pi(E)$ is a loop and $\pi(\omega)\cap\pi(E)$ is not necessarily finite.
• Figure 3: Figure \ref{['fig:i']} illustrates the case when $A$ fulfills condition \ref{['rec:first']} but it falls into case \ref{['caseone']} from the proof and hence is not recurrent. Figure \ref{['fig:ii']} illustrates the case when $A$ is recurrent.

Theorems & Definitions (43)

• Theorem 1.1
• Remark 2.1
• Remark 2.2
• Definition 2.3
• Theorem 2.4: Blokh, B1B2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3