Volume entropy of hyperbolic buildings

TL;DR

This paper characterizes the volume entropy of regular hyperbolic buildings using topological pressure, reveals the Liouville measure is not entropy-maximizing, and provides bounds based on geometric parameters.

Contribution

It introduces a novel characterization of volume entropy in hyperbolic buildings via topological pressure and establishes new bounds related to building geometry.

Findings

Liouville measure is not entropy maximizing

Provides a strict lower bound on volume entropy

Connects entropy with branching numbers and boundary volume

Abstract

We characterize the volume entropy of an arbitrary regular building as the topological pressure of the geodesic flow on an apartment. We show that the Liouville measure is not entropy maximizing measure for regular hyperbolic buildings. As a consequence, we obtain a strict lower bound on the volume entropy in terms of the branching numbers and the volume of the boundary polyhedrons.