Topological Entropy Conjecture

Abstract

In 1974, M. Shub stated Topological Entropy Conjecture, that is, the inequality $\log \rho \le ent(f)$ 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 $f_{\ast }$ which is the linear transformation induced by $f$ on the homology group $H_{\ast }(M;\mathbb{Z})=\underset{i=0}{\overset{n}{⨁}}{H}_i(M;\mathbb{Z})$. In 1986, A. B. Katok gave a counterexample such that the inequality $\log \rho \le ent(f)$ is invalid. In this paper, we define $f$-Čech 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 ${\check{H}}_{\ast }(X;\mathbb{Z})$ exists, where $f\in C^0(X)$ and $J$ is the set of all covers. Then we prove that $\log \rho \le ent(f_L)$ is valid.

15pages