David Jesús Nieves-Rivera

arXiv2021

Integration in Čech theories and a bound on entropy

The evaluation of Alexander-Spanier cochains over formal simplices in a topological space leads to a notion of integration of Alexander-Spanier cohomology classes over Čech homology classes. The integral defines an explicit and non-degenerate pairing between the Alexander-Spanier cohomology and the Čech homology. Instead of working on the limits that define both groups, most of the discussion is carried out "at scale $\mathcal U$", for an open covering $\mathcal U$. As an application, we generalize a result of Manning to arbitrary compact spaces $X$: we prove that the topological entropy of $f \colon X \to X$ is bounded from below by the logarithm of the spectral radius of the map induced in the first Čech cohomology group.

math.AT↗

arXiv2018

Dynamics and eigenvalues in dimension zero

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 $f_{*}:\check{H}_{0}(X;\mathbb{C})\to\check{H}_{0}(X;\mathbb{C})$ 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.

math.DS↗