Exponential dimension: thermodynamic formalism and inversion

Abstract

We present a thermodynamic formalism incorporating the exponential dimension that was proposed to take proper account of the contribution of the harmonic lengths (i.e., lengths that go inversely with n, the stage of refinement of the set of interest). This behavior can be observed in sets such as the Julia set and the irrational windings of the circle map. We obtain the form of the free energy for such sets and find that it has terms proportional to n and lnn, in agreement with the previous numerical estimates for the Julia set. We formulate an inversion procedure capable of exposing the n dependence of the scaling functions of a given set from the knowledge of its exponential and generalized dimensions and show that it gives the correct dependence for known sets.