Pressure, Poincare series and box dimension of the boundary
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Date
2021
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Publisher
IOP PUBLISHING LTD
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 partial differential Hn <i 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.
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Keywords
dimension theory, boundary of hyperbolic space, uniformly expanding Markov maps, MULTIFRACTAL ANALYSIS, GIBBS MEASURES, FINITENESS, MANIFOLDS, ENTROPY