Computation of categorical entropy via spherical functors

TL;DR

This paper investigates the categorical entropy of twist and cotwist functors associated with spherical functors, establishing conditions under which their entropies coincide and applying these findings to a Gromov--Yomdin type conjecture.

Contribution

It proves the equality of categorical entropy for twist and cotwist functors under certain conditions and generalizes previous entropy computations for spherical and P-twists.

Findings

Categorical entropy of twist equals that of cotwist under specific conditions.

Results extend previous entropy calculations for spherical and P-twists.

Application to the Gromov--Yomdin type conjecture demonstrates practical relevance.

Abstract

We study the relationship between the categorical entropy of the twist and cotwist functors along a spherical functor. In particular, we prove the categorical entropy of the twist functor coincides with that of the cotwist functor if the essential image of the right adjoint functor of the spherical functor contains a split-generator. We also see our results generalize the computations of the categorical entropy of spherical twists and $\mathbb{P}$-twists by Ouchi and Fan. As an application, we apply our results to the Gromov--Yomdin type conjecture by Kikuta--Takahashi.