Local Dimensions,Average Densities and Self-Conformal Measures

We consider subsets F of $\mathbb{R}^n$ generated by iterated function systems with contracting conformal C^1+γ-diffeomorphisms whose Hausdorff dimension is s. The unique s-self-conformal probability measure μ agrees with the normalised s-dimensional Hausdorff measure on F. Using the associated dynamical system and ergodic theory we develop a potential theoretic representation of the average densities of μ at almost all points of F. Before we formulate some general relationships between average densities and local dimensions of measures.