[Paper Review] Topological entropy of nonautonomous dynamical systems

Kairan Liu, Yixiao Qiao+1 more|arXiv (Cornell University)|Nov 18, 2019

Mathematical Dynamics and Fractals15 references100 citations

TL;DR

The paper proves that for a nonautonomous dynamical system (NADS), zero topological entropy on X iff zero entropy on its induced system on measures, and positive entropy on X implies infinite entropy on the induced system; it also provides a counterexample showing entropy is not preserved under finite‑to‑one extensions in NADS.

ABSTRACT

Let $\mathcal{M}(X)$ be the space of Borel probability measures on a compact metric space $X$ endowed with the weak$^\ast$-topology. In this paper, we prove that if the topological entropy of a nonautonomous dynamical system $(X,\{f_n\}_{n=1}^{+\infty})$ vanishes, then so does that of its induced system $(\mathcal{M}(X),\{f_n\}_{n=1}^{+\infty})$; moreover, once the topological entropy of $(X,\{f_n\}_{n=1}^{+\infty})$ is positive, that of its induced system $(\mathcal{M}(X),\{f_n\}_{n=1}^{+\infty})$ jumps to infinity. In contrast to Bowen's inequality, we construct a nonautonomous dynamical system whose topological entropy is not preserved under a finite-to-one extension.

Motivation & Objective

• Motivate the study of entropy in nonautonomous dynamical systems (NADS) and their induced measure systems.
• Establish precise entropy relations between a NADS and its induced system on probability measures.
• Investigate whether entropy is preserved under extensions in the nonautonomous setting.
• Provide constructions that illustrate the boundaries of Bowen-type entropy inequalities in NADS.

Proposed method

• Define the induced system on the space of Borel probability measures with the weak* topology.
• Prove h_top(X, {f_n}) = 0 ⇔ h_top(M(X), {f_n}) = 0 for the induced system.
• Show h_top(X, {f_n}) > 0 ⇔ h_top(M(X), {f_n}) = +∞ by embedding X^k into M(X) and using injective, equivariant maps.
• Construct a concrete counterexample demonstrating that a finite‑to‑one extension need not preserve entropy in NADS.
• Utilize a combinatorial lemma (Lemma 4.1) to derive lower bounds on entropy via covering arguments.
• Provide a constructive, explicit finite‑to‑one extension where the entropy inequality h_top(X) > h_top(Y) holds, contrary to Bowen’s inequality in the autonomous case.

Experimental results

Research questions

• RQ1 Does zero entropy of a NADS imply zero entropy of its induced system on measures and vice versa?
• RQ2 What is the relation between positive entropy of a NADS and the entropy of its induced system on measures?
• RQ3 Is entropy preserved under finite‑to‑one extensions for NADS as in the classical (autonomous) setting?
• RQ4 Can explicit counterexamples be constructed to show that Bowen’s inequality fails for NADS?
• RQ5 How does one construct embeddings of X^k into the induced system to obtain lower bounds on entropy?

Key findings

• h_top(X, {f_n}) = 0 if and only if h_top(M(X), {f_n}) = 0.
• h_top(X, {f_n}) > 0 if and only if h_top(M(X), {f_n}) = +∞.
• For every k, X^k with the induced coordinatewise maps has entropy k · h_top(X, {f_n}); hence the induced system has infinite entropy when h_top(X, {f_n}) > 0.
• There exist finite‑to‑one extensions (X, {f_n}) → (Y, {g_n}) for which h_top(X, {f_n}) > h_top(Y, {g_n}), showing Bowen’s entropy inequality does not generally hold for NADS.
• The paper provides a constructive counterexample demonstrating the failure of entropy preservation under finite‑to‑one extensions in the nonautonomous setting.