Topological Entropy Conjecture

TL;DR

This paper investigates the Topological Entropy Conjecture, defining new homology and entropy concepts on compact Hausdorff spaces, and proves the inequality holds in this broader setting.

Contribution

The paper introduces f-Cech homology and topological fiber entropy, establishing the validity of the entropy inequality in a new context.

Findings

Proves $ ext{log} ho \u2264 ent(f_L)$ holds for compact Hausdorff spaces.

Defines new homology and entropy concepts for continuous maps.

Extends the validity of the entropy inequality beyond previous counterexamples.

Abstract

In 1974, M. Shub stated Topological Entropy Conjecture, that is, the inequality $\log\rho\leq ent(f)$ is valid or not, where $f$ is a continuous self-map on a compact manifold $M$, $ent(f)$ is the topological entropy of $f$ and $\rho$ is the maximum absolute eigenvalue of $f_*$ which is the linear transformation induced by $f$ on the homology group $H_{*}(M;\mathbb{Z})=\bigoplus\limits_{i=0}^n{H_{i}(M;\mathbb{Z})}$. In 1986, A. B. Katok gave a counterexample such that the inequality $\log\rho\leq ent(f)$ is invalid. In this paper, we define $f$-\v{C}ech homology group $\check{H}_{i}(X,f;\mathbb{Z})$ and topological fiber entropy $ent(f_L)$ on compact Hausdorff space $X$ for which there is $n=n(J)$ such that $\check{H}_*(X;\mathbb{Z})$ exists, where $f\in C^0(X)$ and $J$ is the set of all covers. Then we prove that $\log\rho\leq ent(f_L)$ is valid.