Topological Entropy Conjecture

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 i...

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Bibliographic Details
Published in:arXiv.org 2018-03
Main Author: Luo, Lvlin
Format: Article
Language:English
Subjects:
Chaos theory
Eigenvalues
Entropy
Homology
Linear transformations
Topology
Online Access:Get full text
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Summary: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 \(f\)-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 \(\check{H}_*(X;\mathbb{Z})\) 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.
ISSN:2331-8422