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Register a new userOn normal numbers and self-similar measures
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Simon Baker
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In this paper we prove that if ${varphi_i(x)=lambda x+t_i}$ is an equicontractive iterated function system and $b$ is a positive integer satisfying $frac{log b}{log |lambda|} otinmathbb{Q},$ then almost every $x$ is normal in base $b$ for any non-atomic self-similar measure of ${varphi_i}$.
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We study random walks on a $d$-dimensional torus by affine expanding maps whose linear parts commute. Assuming an irrationality condition on their translation parts, we prove that the Haar measure is the unique stationary measure. We deduce that if $K subset mathbb{R}^d$ is an attractor of a finite iterated function system of $ngeq 2$ maps of the form $x mapsto D^{-r_i} x + t_i (i=1, ldots, n)$, where $D$ is an expanding $dtimes d$ integer matrix, and is the same for all the maps, and $r_{i} inmathbb{N}$, under an irrationality condition on the translation parts $t_i$, almost every point in $K$ (w.r.t. any Bernoulli measure) has an equidistributed orbit under the map $xmapsto Dx$ (multiplication mod $mathbb{Z}^{d}$). In the one-dimensional case, this conclusion amounts to normality to base $D$. Thus for example, almost every point in an irrational dilation of the middle-thirds Cantor set is normal to base 3.
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A Borel probability measure $mu$ on a locally compact group is called a spectral measure if there exists a subset of continuous group characters which forms an orthogonal basis of the Hilbert space $L^2(mu)$. In this paper, we characterize all spectral measures in the field $mathbb{Q}_p$ of $p$-adic numbers.
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We show that any equicontractive, self-similar measure arising from the IFS of contractions $(S_{j})$, with self-similar set $[0,1]$, admits an isolated point in its set of local dimensions provided the images of $S_{j}(0,1)$ (suitably) overlap and the minimal probability is associated with one (resp., both) of the endpoint contractions. Examples include $m$-fold convolution products of Bernoulli convolutions or Cantor measures with contraction factor exceeding $1/(m+1)$ in the biased case and $1/m$ in the unbiased case. We also obtain upper and lower bounds on the set of local dimensions for various Bernoulli convolutions.
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