Slow Entropy of Some Parabolic Flows

TL;DR

This paper investigates slow entropy invariants of quasi-unipotent parabolic flows on homogeneous spaces, showing how to compute topological complexity from algebraic structure and establishing a variational principle that links metric and topological entropy.

Contribution

It introduces a method to compute slow entropy from Jordan block structures and proves the variational principle for quasi-unipotent flows, including non-compact cases.

Findings

Topological complexity can be derived from Jordan block structure.

Metric and topological slow entropy coincide for these flows.

Examples show measure-theoretic and topological complexities can differ.

Abstract

We study nontrivial entropy invariants in the class of parabolic flows on homogeneous spaces, quasi-unipotent flows. We show that topological complexity (ie, slow entropy) can be computed directly from the Jordan block structure of the adjoint representation. Moreover using uniform polynomial shearing we are able to show that the metric orbit growth (ie, slow entropy) coincides with the topological one, establishing hence variational principle for quasi-unipotent flows (this also applies to the non-compact case). Our results also apply to sequence entropy. We establish criterion for a system to have trivial topological complexity and give some examples in which the measure-theoretic and topological complexities do not coincide for uniquely ergodic systems, violating the intuition of the classical variational principle.