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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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It is known that the heuristic principle, referred to as the multifractal formalism, need not hold for self-similar measures with overlap, such as the $3$-fold convolution of the Cantor measure and certain Bernoulli convolutions. In this paper we study an important function in the multifractal theory, the $L^{q}$-spectrum, $tau (q)$, for measures of finite type, a class of self-similar measures that includes these examples. Corresponding to each measure, we introduce finitely many variants on the $% L^{q}$-spectrum which arise naturally from the finite type structure and are often easier to understand than $tau $. We show that $tau$ is always bounded by the minimum of these variants and is equal to the minimum variant for $qgeq 0$. This particular variant coincides with the $L^{q}$-spectrum of the measure $mu$ restricted to appropriate subsets of its support. If the IFS satisfies particular structural properties, which do hold for the above examples, then $tau$ is shown to be the minimum of these variants for all $q$. Under certain assumptions on the local dimensions of $mu$, we prove that the minimum variant for $q ll 0$ coincides with the straight line having slope equal to the maximum local dimension of $mu $. Again, this is the case with the examples above. More generally, bounds are given for $tau$ and its variants in terms of notions closely related to the local dimensions of $mu $.
The upper and lower Assouad dimensions of a metric space are local variants of the box dimensions of the space and provide quantitative information about the `thickest and `thinnest parts of the set. Less extre
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 ${varphi_i}$.
S. Baker (2019), B. Barany and A. K{a}enm{a}ki (2019) independently showed that there exist iterated function systems without exact overlaps and there are super-exponentially close cylinders at all small levels. We adapt the method of S. Baker and obtain further examples of this type. We prove that for any algebraic number $betage 2$ there exist real numbers $s, t$ such that the iterated function system $$ left {frac{x}{beta}, frac{x+1}{beta}, frac{x+s}{beta}, frac{x+t}{beta}right } $$ satisfies the above property.
We investigate separation properties of $N$-point configurations that minimize discrete Riesz $s$-energy on a compact set $Asubset mathbb{R}^p$. When $A$ is a smooth $(p-1)$-dimensional manifold without boundary and $sin [p-2, p-1)$, we prove that the order of separation (as $Nto infty$) is the best possible. The same conclusions hold for the points that are a fixed positive distance from the boundary of $A$ whenever $A$ is any $p$-dimensional set. These estimates extend a result of Dahlberg for certain smooth $(p-1)$-dimensional surfaces when $s=p-2$ (the harmonic case). Furthermore, we obtain the same separation results for `greedy $s$-energy points. We deduce our results from an upper regularity property of the $s$-equilibrium measure (i.e., the measure that solves the continuous minimal Riesz $s$-energy problem), and we show that this property holds under a local smoothness assumption on the set $A$.
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