Dimension of self-conformal measures associated to an exponentially separated holomorphic IFS

Abstract

Let $\Phi$ be a holomorphic IFS on a bounded domain in $\mathbb{C}$. Suppose that the following conditions hold: (1) the maps in $\Phi$ do not have a common fixed point; (2) there does not exist a regular real-analytic curve which is invariant under all of the maps in $\Phi$; (3) $\Phi$ is not holomorphically conjugate to a homothetic IFS; (4) $\Phi$ is exponentially separated. Under these assumptions, we show that the dimensions of the self-conformal measures associated to $\Phi$, as well as the Hausdorff dimension of the associated self-conformal set, attain their natural upper bounds. The proof combines recently developed methods from the dimension theory of stationary fractal measures with complex-analytic arguments.

41 pages