Ergodic limits on the conformal repellers
Let \(M\) be a smooth Riemannian manifold and \(g:M\rightarrow M\) be a \(C^{1+\delta}\)-conformal map. Let \(\Phi\) be a continuous function defined on a \(g\)-invariant compact subset \(J\) of \(M\) with values in \(\mathbb{R}^{d}.\) For any \(x\in J,\) the ergodic limit, if exists, is defined by \[ \alpha(x)=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{j=0}^{n-1}\Phi(g^{j}x). \] For \(\alpha\in L_{\Phi},\) let \[ E(\alpha)=\left\{ x\in J:\alpha(x)=\alpha\right\} , \] where \[ L_{\Phi}=\left\{ \alpha : \alpha=\alpha(x)\text{ for some }x\in J\right\} . \] The main purpose of the paper is to examine the size of the set \(E(\alpha)\) as well as the size of the set of points such that the limits defining \(\alpha(x)\) do not exist. The sizes of these sets are described in terms of the Hausdorff dimension \(\dim_{H}\) and the packing dimension \(\dim_{P}.\)
scientific article; zbMATH DE number 1800237
scientific article