Critical base for the unique codings of fat Sierpinski gasket

Abstract

Given $\beta \in (1,2)$ the fat Sierpinski gasket ${\mathcal{S}}_{\beta }$ is the self-similar set in ${\mathbb{R}}^2$ generated by the iterated function system (IFS) \[ f_{β,d}(x)=\frac{x+d}β,\quad d\in\mathcal A:=\{(0, 0), (1,0), (0,1)\}. \] Then for each point $P\in {\mathcal{S}}_{\beta }$ there exists a sequence $(d_i)\in {\mathcal{A}}^{\mathbb{N}}$ such that $P={\sum }_{i=1}^{\infty }{d}_i/{\beta }^i$, and the infinite sequence $(d_i)$ is called a \emph{coding} of $P$. In general, a point in ${\mathcal{S}}_{\beta }$ may have multiple codings since the overlap region ${\mathcal{O}}_{\beta }:={\bigcup }_{c,d\in \mathcal{A},c\ne d}{f}_{\beta ,c}({\Delta }_{\beta })\cap f_{\beta ,d}({\Delta }_{\beta })$ has non-empty interior, where ${\Delta }_{\beta }$ is the convex hull of ${\mathcal{S}}_{\beta }$. In this paper we are interested in the invariant set \[ \widetilde{\mathcal U}_β:=\left\{\sum_{i=1}^\infty \frac{d_i}{β^i}\in \mathcal S_β: \sum_{i=1}^\infty\frac{d_{n+i}}{β^i}\notin\mathcal O_β~\forall n\ge 0\right\}. \] Then each point in ${\tilde{\mathcal{U}}}_{\beta }$ has a unique coding. We show that there is a transcendental number ${\beta }_c\approx 1.55263$ related to the Thue-Morse sequence, such that ${\tilde{\mathcal{U}}}_{\beta }$ has positive Hausdorff dimension if and only if $\beta >{\beta }_c$. Furthermore, for $\beta ={\beta }_c$ the set ${\tilde{\mathcal{U}}}_{\beta }$ is uncountable but has zero Hausdorff dimension, and for $\beta <{\beta }_c$ the set ${\tilde{\mathcal{U}}}_{\beta }$ is at most countable. Consequently, we also answer a conjecture of Sidorov (2007). Our strategy is using combinatorics on words based on the lexicographical characterization of ${\tilde{\mathcal{U}}}_{\beta }$.

28 pages, 10 figures

References in corpus (4)

• Hausdorff dimension of unique beta expansions
• Greedy and quasi-greedy expansions in non-integer bases
• Entropy, topological transitivity, and dimensional properties of unique $q$-expansions
• Bifurcation sets arising from non-integer base expansions

Cited by in corpus (1)

• Intersections of Siepinski gasket with its translation