Topological Entropy Conjecture

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.