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

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.

23 pages. Comments are welcome