The paper analyses the normalized Hausdorff measure on the fractal set arising from backward trajectories of a conformal mapping of the Euclidean space. Finitely many conformal diffeomorphisms are considered satisfying the strong open set condition and having Hölder continuous contraction ratios. A normalized \(s\)-dimensional Hausdorff measure \(\mu \) lives on the associated self-conformal fractal set. The paper focusses in the average \(s\)-density of \(\mu \), in relation with the potential theoretic interpretation of the Hausdorff dimension as the capacity invariant.
The main theorem of the paper states that the average density of \(\mu \) equals a positive constant given by a truncated generalized \(s\)-energy integral. A corollary also provides an approximation formula appropriate to numerical simulation.
The main initial strategy of the proof is the construction and analysis of a conditional Gibbs measure. We remark the quite general and independent interest of the final tool of the proof, consisting of an extension of Birkhoff’s ergodic theorem, that is stated and proved independently in the appendix of this article.