Pressure, Poincar\'e series and box dimension of the boundary

TL;DR

This paper links the box dimension of boundaries and iterates in hyperbolic spaces to pressure and Poincaré series, revealing deep connections between dynamical systems, geometric group theory, and dimension theory.

Contribution

It establishes new relationships between box dimension, pressure, and Poincaré series for certain dynamical and geometric group actions.

Findings

Box dimension of boundary relates to pressure of geometric potential.

Box dimension of iterates in hyperbolic space equals the critical exponent of Poincaré series.

Connects entropy at infinity with dimension theory.

Abstract

In this note we prove two related results. First, we show that for certain Markov interval maps with infinitely many branches the upper box dimension of the boundary can be read from the pressure of the geometric potential. Secondly, we prove that the box dimension of the set of iterates of a point in H^n with respect to a parabolic subgroup of isometries equals the critical exponent of the Poincare series of the associated group. This establishes a relationship between the entropy at infinity and dimension theory.