Published online by Cambridge University Press: 04 January 2019
Let $X$ be a compact, metric and totally disconnected space and let
$f:X\rightarrow X$ be a continuous map. We relate the eigenvalues of
$f_{\ast }:\check{H}_{0}(X;\mathbb{C})\rightarrow \check{H}_{0}(X;\mathbb{C})$ 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 a classical inequality of Manning that bounds the entropy of
$f$ below by the spectral radius of
$f_{\ast }$.