On topological entropy, Lefschetz numbers and Lefschetz zeta functions

Llibre, Jaume; Sirvent, Vı́ctor F.

doi:10.1016/j.topol.2019.106906

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2021

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“…Lately, for results about random entropy expansiveness and dominated splittings, see [38], and for results about the relations of topological entropy and Lefschetz numbers, see [39][40][41]. Furthermore, for a variational principle for subadditive preimage topological pressure for continuous bundle random dynamical systems, see [42].…”

Section: Brief Historymentioning

confidence: 99%

The Topological Entropy Conjecture

Luo

2021

Mathematics

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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 nJ, where nJ is the dimension of X associated with ∂. Therefore, we have Hˇp(X;Z), where 0≤p≤n=nJ. For a continuous self-map f on X, let α∈J be an open cover of X and Lf(α)={Lf(U)|U∈α}. Then, there exists an open fiber cover L˙f(α) of Xf induced by Lf(α). In this paper, we define a topological fiber entropy entL(f) as the supremum of ent(f,L˙f(α)) through all finite open covers of Xf={Lf(U);U⊂X}, where Lf(U) is the f-fiber of U, that is the set of images fn(U) and preimages f−n(U) for n∈N. Then, we prove the conjecture logρ≤entL(f) for f being a continuous self-map on a given compact Hausdorff space X, where ρ is the maximum absolute eigenvalue of f*, which is the linear transformation associated with f on the Čech homology group Hˇ*(X;Z)=⨁i=0nHˇi(X;Z).

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Section: Brief Historymentioning

confidence: 99%

The Topological Entropy Conjecture

Luo

2021

Mathematics

View full text Add to dashboard Cite

show abstract

On topological entropy, Lefschetz numbers and Lefschetz zeta functions

Cited by 1 publication

References 20 publications

The Topological Entropy Conjecture

The Topological Entropy Conjecture

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