{"paper":{"title":"Topological Entropy Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.DS","authors_text":"Lvlin Luo","submitted_at":"2012-09-18T12:51:45Z","abstract_excerpt":"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;\\mathb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3936","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}