Random β-transformation on fat Sierpinski gasket

Abstract

We consider the iterated function system (IFS) $$f_{\vec{q}}(\vec{z})=\frac{\vec{z}+\vec{q}}{\beta },\vec{q}\in \{(0,0),(1,0),(0,1)\}.$$ As is well known, for $\beta =2$ the attractor, $S_{\beta }$, is a fractal called the Sierpiński gasket(or sieve) and for $\beta >2$ it is also a fractal. Our goal is to study greedy, lazy and random $\beta$-transformations on the attractor for this IFS with $1<\beta <2$. For $1<\beta \le 3/2$, $S_{\beta }$ is a triangle and it is shown that the greedy transformation $T_{\beta }$ and the lazy transformation $L_{\beta }$ are isomorphic and they both admit an absolutely continuous invariant measure. We show that all $\beta$-expansions of a point $\vec{z}$ in $S_{\beta }$ can be generated by a random map $K_{\beta }$ defined on $\{0,1{\}}^{\mathbb{N}}\times \{0,1,2{\}}^{\mathbb{N}}\times S_{\beta }$ and $K_{\beta }$ has a unique invariant measure of maximal entropy when $1<\beta \le {\beta }_{\ast }$, where ${\beta }_{\ast }\approx 1.4656$ is the root of $x^3-x^2-1=0$. We also show existence of a $K_{\beta }$-invariant probability measure, absolutely continuous with respect to $m_1\otimes m_2\otimes {\lambda }_2$, where $m_1,m_2$ are product measures on $\{0,1{\}}^{\mathbb{N}},\{0,1,2{\}}^{\mathbb{N}}$, respectively, and ${\lambda }_2$ is the normalized Lebesgue measure on $S_{\beta }$. For $3/2<\beta \le {\beta }^{\ast }$, where ${\beta }^{\ast }\approx 1.5437$ is the root of $x^3-2x^2+2x=2$, there are radial holes in $S_{\beta }$. In this case, $K_{\beta }$ is defined on $\{0,1{\}}^{\mathbb{N}}\times S_{\beta }$. We also show that it has a unique invariant measure of maximal entropy.