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

TL;DR

This paper proves a lower bound estimate for the topological entropy of continuous self-maps on certain compact manifolds, confirming the Entropy Conjecture for infra-nilmanifolds and related spaces.

Contribution

It extends previous results to confirm the Entropy Conjecture for a broader class of manifolds and provides a new lower estimate of entropy using spectral radius of associated matrices.

Findings

Confirmed the Entropy Conjecture for continuous maps on infra-nilmanifolds.

Established a lower bound for topological entropy based on spectral radius.

Derived absolute lower bounds for entropy using Mahler measure estimates.

Abstract

Extending our results in "Entropy conjecture for continuous maps of nilmanifolds", to appear in Israel Jour. of Math., 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 logarithm of the spectral radius 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.