Phase Transitions on Strange Sets

Abstract

Abstract:The "thermodynamic" formalism provides a very general division of strange (Cantor, fractal) sets into two classes; those which do exhibit phase transitions, and those which do not.The "thermodynamic formalism,,1,2,3,4) is based on the observation that the sum used in determining the Hausdorff dimension of a strange (fractal, Cantor) set resembles a partition sum over "configurations" i, with T playing the role of "temperature", and covering interval sizes playing the role of "Boltzmann weights".The "thermodynamic" functions extracted from the above partition sum exhibit a phenomenon that might have gone unnoticed were it not for the thermodynamic formalism. They can undergo "phase transitions". These phase transitions can be visualized in the following way: the exponent T acts as a "magnifying lens" which blows up some of the covering intervals, and (relatively), shrinks the others. For negative T, the fat intervals are expanded, and the thin ones are made even thinner. For positive T, the thin intervals are (relatively) blown up. Below the phase transition, the sum is dominated by a few fat intervals. Visually, the cover consists of a few + Carlsberg Fellow