Categorical entropy, (co-)t-structures and ST-triples

TL;DR

This paper introduces new entropy invariants for exact endofunctors on triangulated categories with t-structures and co-t-structures, exploring how these invariants relate within subcategories and their dynamical properties.

Contribution

It develops new entropy-type invariants using bounded (co-)t-structures and applies them to relate categorical entropies in dual subcategories.

Findings

Introduced entropy invariants based on bounded (co-)t-structures.

Proved properties of these invariants in triangulated categories.

Applied invariants to relate entropies in dual subcategories.

Abstract

In this paper, we study a dynamical property of an exact endofunctor $\Phi : \mathcal{D} \to \mathcal{D}$ of a triangulated category $\mathcal{D}$. In particular, we are interested in the following question: Given full triangulated subcategories $\mathcal{A},\mathcal{B} \subset \mathcal{D}$ such that $\Phi(\mathcal{A}) \subset \mathcal{A}$ and $\Phi(\mathcal{B}) \subset \mathcal{B}$, how the categorical entropies of $\Phi|_\mathcal{A}$ and $\Phi|_\mathcal{B}$ are related? To answer this question, we introduce new entropy-type invariants using bounded (co-)t-structures with finite (co-)hearts and prove their basic properties. We then apply these results to answer our question for the situation where $\mathcal{A}$ has a bounded t-structure and $\mathcal{B}$ has a bounded co-t-structure which are, in some sense, dual to each other.