Estimates of the topological entropy from below for continuous self-maps on some compact manifolds

Wacław Marzantowicz; Feliks Przytycki

doi:10.3934/dcds.2008.21.501

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2008, Volume 21, Issue 2: 501-512. Doi: 10.3934/dcds.2008.21.501

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Estimates of the topological entropy from below for continuous self-maps on some compact manifolds

Wacław Marzantowicz^1, and
Feliks Przytycki^2,

1.
Faculty of Mathematics and Comp. Sci., Adam Mickiewicz University of Poznań, ul. Umultowska 87, 61-614 Poznań
2.
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-950 Warszawa

Received: March 31, 2007

Revised: September 30, 2007

Published: April 2008

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Abstract

Extending our results of [17], we confirm that Entropy Conjecture holds for every continuous self-map of a compact $K(\pi,1)$ manifold with the fundamental group $\pi$ torsion free and virtually nilpotent, in particular for every continuous map of an infra-nilmanifold. In fact we prove a stronger version, a lower estimate of the topological entropy of a map by the logarithm of the spectral radius of exterior power of an associated "linearization matrix" with integer entries.
From this, referring to known estimates of Mahler measure of polynomials, we deduce some absolute lower bounds for the entropy.

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Mathematics Subject Classification: Primary: 37B40; Secondary: 22E25, 37D20, 57T15.

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