Projections of cartesian products of the self-similar sets without the irrationality assumption

Kan Jiang

paper

Projections of cartesian products of the self-similar sets without the irrationality assumption

arXiv:1806.01080

Abstract

Let $β > 1$ . Define a class of similitudes \[S=\left\{f_{i}(x)=\dfrac{x}{β^{n_i}}+a_i:n_i\in \mathbb{N}^{+}, a_i\in \mathbb{R}\right\}.\] Let $A$ be the collection of all the self-similar sets generated by the similitudes from $S$ . In this paper, we prove that for any $θ \in (0, π)$ and $K_{1}, K_{2} \in A$ , $P r o j_{θ} (K_{1} \times K_{2})$ is similar to a self-similar set or an attractor of some infinite iterated function system, where $P r o j_{θ}$ denotes the orthogonal projection onto $L_{θ}$ , and $L_{θ}$ denotes the line through the origin in direction $θ$ . As a corollary, $dim_{P} (P r o j_{θ} (K_{1} \times K_{2})) = dim_{B} (P r o j_{θ} (K_{1} \times K_{2}))$ holds for any $θ \in (0, π)$ and any $K_{1}, K_{2} \in A$ , where $dim_{P}$ and $dim_{B}$ denote the packing and upper box dimension. Whether $P r o j_{θ} (K_{1} \times K_{2})$ is similar to a self-similar set or not is uniquely determined by the similarity ratios of $K_{1}$ and $K_{2}$ rather than the angle $θ .$ When $P r o j_{θ} (K_{1} \times K_{2})$ is similar to a self-similar set, in terms of the finite type condition \cite{NW}, we are able to calculate in cerntain cases the Hausdorff dimension of $P r o j_{θ} (K_{1} \times K_{2})$ . If $P r o j_{θ} (K_{1} \times K_{2})$ is similar to an attractor of some infinite iterated function system, then by virtue of the Vitali covering lemma \cite{FG} we give an estimation of the Hausdorff dimension of $P r o j_{θ} (K_{1} \times K_{2})$ . For some cases, we can calculate, by means of Mauldin and Urbanski' result \cite{MRD}, the exact Hausdorff dimension of $P r o j_{θ} (K_{1} \times K_{2})$ . We also find some non-trivial examples such that for some angle $θ \in [0, π)$ and some $K_{1}, K_{2} \in A$ , $dim_{H} (P r o j_{θ} (K_{1} \times K_{2})) = dim_{H} (K_{1}) + dim_{H} (K_{2})$ .

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Projections of cartesian products of the self-similar sets without the irrationality assumption

Abstract

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