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- W2897360291 abstract "Let $F={mathbf{p}_0,ldots,mathbf{p}_n}$ be a collection of points in $mathbb{R}^d.$ The set $F$ naturally gives rise to a family of iterated function systems consisting of contractions of the form $$S_i(mathbf{x})=lambda mathbf{x} +(1-lambda)mathbf{p}_i,$$ where $lambda in(0,1)$. Given $F$ and $lambda$ it is well known that there exists a unique non-empty compact set $X$ satisfying $X=cup_{i=0}^n S_i(X)$. For each $mathbf{x} in X$ there exists a sequence $mathbf{a}in{0,ldots,n}^{mathbb{N}}$ satisfying $$mathbf{x}=lim_{jtoinfty}(S_{a_1}circ cdots circ S_{a_j})(mathbf{0}).$$ We call such a sequence a coding of $mathbf{x}$. In this paper we prove that for any $F$ and $k inmathbb{N},$ there exists $delta_k(F)>0$ such that if $lambdain(1-delta_k(F),1),$ then every point in the interior of $X$ has a coding which is $k$-simply normal. Similarly, we prove that there exists $delta_{uni}(F)>0$ such that if $lambdain(1-delta_{uni}(F),1),$ then every point in the interior of $X$ has a coding containing all finite words. For some specific choices of $F$ we obtain lower bounds for $delta_k(F)$ and $delta_{uni}(F)$. We also prove some weaker statements that hold in the more general setting when the similarities in our iterated function systems exhibit different rates of contraction. Our proofs rely on a variation of a well known construction of a normal number due to Champernowne, and an approach introduced by ErdH{o}s and Komornik." @default.
- W2897360291 created "2018-10-26" @default.
- W2897360291 creator A5056602178 @default.
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- W2897360291 date "2018-10-16" @default.
- W2897360291 modified "2023-09-26" @default.
- W2897360291 title "On the complexity of the set of codings for self-similar sets and a variation on the construction of Champernowne" @default.
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- W2897360291 doi "https://doi.org/10.48550/arxiv.1810.07083" @default.
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