A comparison of categorical and topological entropies on Weinstein manifolds

TL;DR

This paper establishes a lower bound for the topological entropy of symplectic automorphisms on Weinstein manifolds using categorical entropy, and proposes a conjecture linking these concepts in dynamical systems.

Contribution

It proves that categorical entropy bounds topological entropy from below on Weinstein manifolds and introduces a conjecture generalizing a dynamical systems result.

Findings

Categorical entropy provides a lower bound for topological entropy.

The work extends entropy comparisons to Weinstein manifolds.

A new conjecture relating categorical and topological entropy is proposed.

Abstract

Let $W$ be a symplectic manifold, and let $\phi:W \to W$ be a symplectic automorphism. Then, $\phi$ induces an auto-equivalence $\Phi$ defined on the Fukaya category of $W$. In this paper, we prove that the categorical entropy of $\Phi$ bounds the topological entropy of $\phi$ from below where $W$ is a Weinstein manifold and $\phi$ is compactly supported. Moreover, being motivated by the work of Cineli, Ginzburg, and Gurel, we propose a conjecture which generalizes a result in dynamical system.