Categorical vs topological entropy of autoequivalences of surfaces

TL;DR

This paper explores the relationship between categorical and topological entropy of autoequivalences on surfaces, providing examples and counterexamples, and analyzing their actions on cohomology.

Contribution

It presents the first example of an autoequivalence with positive categorical entropy on surfaces with a (-2)-curve and relates spectral radii to topological entropy.

Findings

Constructed an autoequivalence with positive categorical entropy.

Provided a counter-example to Kikuta and Takahashi's conjecture.

Linked spectral radii of autoequivalences to topological entropy.

Abstract

In this paper, we give an example of an autoequivalence with positive categorical entropy (in the sense of Dimitrov, Haiden, Katzarkov and Kontsevich) for any surface containing a (-2)-curve. Then we show that this equivalence gives another counter-example to a conjecture proposed by Kikuta and Takahashi. In a second part, we study the action on cohomology induced by spherical twists composed with standard autoequivalences on a surface S and show that their spectral radii correspond to the topological entropy of the corresponding automorphisms of S.