Dynamics and eigenvalues in dimension zero

Abstract

Let $X$ be a compact, metric and totally disconnected space and let $f:X\to X$ be a continuos map. We relate the eigenvalues of $f_{\ast }:{\check{H}}_0(X;\mathbb{C})\to {\check{H}}_0(X;\mathbb{C})$ to dynamical properties of $f$, roughly showing that if the dynamics is complicated then every complex number of modulus different from 0,1 is an eigenvalue. This stands in contrast with the classical Manning's inequality.

Version accepted for publication in Ergod. Theory Dyn. Syst