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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. 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