Local dimension of self similar measures

Kevin Hare (University of Waterloo)

Frank Adams 1,

Let \(S_0, S_1 \dots, S_k\) be a finite set of linear contractions, and \(p_0, p_1, \dots, p_k > 0\) be probabilites with \(p_0 + p_1 + \dots + p_k = 1\). There exists a unique self-similar measure \(\mu\) such that
\(\begin{equation*}
\mu= p_0 \mu \circ S_{0}^{-1} + \dots +
+ p_k \mu \circ S_{k}^{-1}.
\end{equation*}\)
We call such a \(\mu\) a self-similar measure. We define the local dimension of \(\mu\) at \(x\) as
\(\begin{equation*}
\dim_{\mathrm{loc}}\mu_\beta (x)=\lim_{r\rightarrow 0^{+}}\frac{\log \mu_\beta
([x-r,x+r])}{\log r}.
\end{equation*}\)
In this talk we will discuss how one can compute the set of all possible local dimensions for a self-similar measure of finite type.

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