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Dynamics and eigenvalues in dimension zero
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 40 / Issue 9 / September 2020
- Published online by Cambridge University Press:
- 04 January 2019, pp. 2434-2452
- Print publication:
- September 2020
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- Article
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- You have access
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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 
$f_{\ast }$.
