NORMAL AND NON-NORMAL POINTS OF SELF-SIMILAR SETS AND DIVERGENCE POINTS OF SELF-SIMILAR MEASURES

Abstract

Let $K$ and $\mu$ be the self-similar set and the self-similar measure associated with an IFS (iterated function system) with probabilities $(S_i, p_i)_{i=1,\ldots,N}$ satisfying the open set condition. Let $\Sigma=\{1,\ldots,N\}^{\bb N}$ denote the full shift space and let $\pi : \Sigma \longrightarrow K$ denote the natural projection. The (symbolic) local dimension of $\mu$ at $\omega \in \Sigma$ is defined by $\lim_n (\log \mu K_{\omega\mid n}/\log\hbox{ diam }K_{\omega\mid n})$ , where $K_{\omega\mid n}=S_{\omega 1}\circ\ldots\circ S_{\omega_n}(K)$ for $\omega = (\omega_1, \omega_2,\ldots) \in \Sigma$ . A point $\omega$ for which the limit $\lim_n (\log \mu K_{\omega\mid n}/\log\hbox{ diam }K_{\omega\mid n})$ does not exist is called a divergence point. In almost all of the literature the limit $\lim_n (\log \mu K_{\omega\mid n}/\log\hbox{ diam }K_{\omega\mid n})$ is assumed to exist, and almost nothing is known about the set of divergence points. In the paper a detailed analysis is performed of the set of divergence points and it is shown that it has a surprisingly rich structure. For a sequence $(x_n)_n$ , let ${\sf A}(x_n)$ denote the set of accumulation points of $(x_n)_n$ . For an arbitrary subset $I$ of ${\bb R}$ , the Hausdorff and packing dimension of the set

\[ \left\{\omega\in\Sigma\left\vert {\sf A}\left(\frac{\log\mu K_{\omega\mid n}}{\log\hbox{ diam }K_{\omega\mid n}}\right)\right.=I\right\} \]

and related sets is computed. An interesting and surprising corollary to this result is that the set of divergence points is extremely ‘visible’; it can be partitioned into an uncountable family of pairwise disjoint sets each with full dimension.

In order to prove the above statements the theory of normal and non-normal points of a self-similar set is formulated and developed in detail. This theory extends the notion of normal and non-normal numbers to the setting of self-similar sets and has numerous applications to the study of the local properties of self-similar measures including a detailed study of the set of divergence points.