### Dimension theory of iterated function systemsDimension theory of iterated function systems

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 Author Feng, De-Jun ♦ Hu, Huyi Source arXiv.org Content type Text File Format PDF Date of Submission 2010-02-10 Language English
 Subject Domain (in DDC) Natural sciences & mathematics ♦ Mathematics Subject Keyword Mathematics - Dynamical Systems ♦ Mathematics - Classical Analysis and ODEs ♦ 28A78 ♦ 37C45 ♦ 37A45 ♦ 28A80 ♦ 11Z05 ♦ math Abstract Let $\{S_i\}_{i=1}^\ell$ be an iterated function system (IFS) on $\R^d$ with attractor $K$. Let $(\Sigma,\sigma)$ denote the one-sided full shift over the alphabet $\{1,..., \ell\}$. We define the projection entropy function $h_\pi$ on the space of invariant measures on $\Sigma$ associated with the coding map $\pi: \Sigma\to K$, and develop some basic ergodic properties about it. This concept turns out to be crucial in the study of dimensional properties of invariant measures on $K$. We show that for any conformal IFS (resp., the direct product of finitely many conformal IFS), without any separation condition, the projection of an ergodic measure under $\pi$ is always exactly dimensional and, its Hausdorff dimension can be represented as the ratio of its projection entropy to its Lyapunov exponent (resp., the linear combination of projection entropies associated with several coding maps). Furthermore, for any conformal IFS and certain affine IFS, we prove a variational principle between the Hausdorff dimension of the attractors and that of projections of ergodic measures. Description Reference: Comm. Pure Appl. Math., 62 (2009), no. 11, 1435--1500 Educational Use Research Learning Resource Type Article Page Count 60