## Local dimension of self similar measures

#### Kevin Hare (University of Waterloo)

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
$$\begin{equation*} \dim_{\mathrm{loc}}\mu_\beta (x)=\lim_{r\rightarrow 0^{+}}\frac{\log \mu_\beta ([x-r,x+r])}{\log r}. \end{equation*}$$