Explicit formulas for the average density of conformal repellers

Abstract

We obtain explicit formulas for the average density, also called order-two density, of an arbitrary conformal repeller of a $C^{1+\epsilon}$ transformation. We note that unlike with the pointwise densities, the average densities exist almost everywhere on each repeller, and are in fact constant almost everywhere. Thus, they are natural parameters to describe the geometric structure of the corresponding invariant sets, and as such it is of interest to obtain explicit formulas for their almost constant value. We obtain simultaneously formulas without using symbolic dynamics, and formulas based on the symbolic dynamics generated by a given Markov partition. The proofs strongly depend on the use of appropriate Markov partitions, and are also based on methods of Falconer and of Zähle.