Thermodynamic Formalism and Multifractal Analysis of Conformal Infinite Iterated Function Systems

Abstract

We develop the thermodynamic formalism for equilibrium states of strongly Holder families of functions. These equilibrium states are supported on the limit set generated by iterating a system of infinitely many contractions. The theory of these systems was laid out in an earlier paper of the last two authors. The first five sections of this paper except Section 3 are devoted to developing the thermodynamic formalism for equilibrium states of Holder families of functions. The first three sections provide us with the tools needed to carry out the multifractal analysis for the equilibrium states mentioned above assuming that the limit set is generated by conformal contractions. The theory of infinite systems of conformal contractions is laid out in [13]. The multifractal analysis is then given in Section 7. In Section 8 we apply this theory to some examples from continued fraction systems and Apollonian packing.