Pressure, Poincaré series and box dimension of the boundary

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 with respect to a parabolic subgroup of isometries equals the critical exponent of the Poincaré series of the associated group. This establishes a relationship between the entropy at infinity and dimension theory.