Dynamical sheaves

TL;DR

This paper develops a topos-theoretic framework for classifying sites with semigroup actions, establishing equivalences with sheaves with actions, and applies it to Holomorphic Dynamics to interpret key inequalities and rigidity results.

Contribution

It introduces a new classifying site for semigroup actions on small sites and connects it with sheaves with actions, providing tools for applications in Holomorphic Dynamics.

Findings

Established an equivalence between the classifying topos and sheaves with semigroup action.

Derived a spectral sequence for sheaf cohomology in the classifying site.

Applied the formalism to interpret results in Holomorphic Dynamics, such as the Fatou-Shishikura Inequality.

Abstract

In the present work we define and study the classifying (or "quotient") site $[X/\Sigma]$ for any small site $X$ with (countable) coproducts endowed with an action of a (countable) semigroup $\Sigma$. A simple case (the most relevant to our applications) is the case $\Sigma=\mathbb{N}$, on which, therefore we concentrate. Our main result consists in establishing an equivalence of the corresponding T\`opos with the category of sheaves on $X$ with ``$\Sigma-$action''. We prove also that there is a spectral sequence computing sheaf cohomology in $[X/\mathbb{N}]$ and we deduce some topological properties of this site, such as its fundamental group. We finally apply the above formalism in Holomorphic Dynamics, giving a T\`opos-theoretic interpretation of Epstein's work on the Fatou-Shishikura Inequality and Infinitesimal Thurston's Rigidity.