The Topological Entropy Conjecture

Lvlin Luo
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Lvlin Luo: Arts and Sciences Teaching Department, Shanghai University of Medicine and Health Sciences, Shanghai 201318, China

Mathematics, 2021, vol. 9, issue 4, 1-17

Abstract: For a compact Hausdorff space X , let J be the ordered set associated with the set of all finite open covers of X such that there exists n J , where n J is the dimension of X associated with ? . Therefore, we have H ? p ( X ; Z ) , where 0 ? p ? n = n J . For a continuous self-map f on X , let ? ? J be an open cover of X and L f ( ? ) = { L f ( U ) | U ? ? } . Then, there exists an open fiber cover L ? f ( ? ) of X f induced by L f ( ? ) . In this paper, we define a topological fiber entropy e n t L ( f ) as the supremum of e n t ( f , L ? f ( ? ) ) through all finite open covers of X f = { L f ( U ) ; U ? X } , where L f ( U ) is the f-fiber of U , that is the set of images f n ( U ) and preimages f ? n ( U ) for n ? N . Then, we prove the conjecture log ? ? e n t L ( f ) for f being a continuous self-map on a given compact Hausdorff space X , where ? is the maximum absolute eigenvalue of f * , which is the linear transformation associated with f on the ?ech homology group H ? * ( X ; Z ) = ? i = 0 n H ? i ( X ; Z ) .

Keywords: algebra equation; ?ech homology group; ?ech homology germ; eigenvalue; topological fiber entropy (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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