Random β-transformation on fat Sierpinski gasket

We consider the iterated function system (IFS) $$f_{\vec{q}}(\vec{z})=\frac{\vec{z}+\vec{q}}β,\vec{q}\in\{(0,0),(1,0),(0,1)\}.$$ As is well known, for $β= 2$ the attractor, $S_β$, is a fractal called the Sierpiński gasket(or sieve) and for $β>2$ it is also a fractal. Our goal is to study greedy, lazy and random $β$-transformations on the attractor for this IFS with $1<β<2$. For $1<β\leq 3/2$, $S_β$ is a triangle and it is shown that the greedy transformation $T_β$ and the lazy transformation $L_β$ are isomorphic and they both admit an absolutely continuous invariant measure. We show that all $β$-expansions of a point $\vec{z}$ in $S_β$ can be generated by a random map $K_β$ defined on $\{0,1\}^\mathbb{N}\times\{0,1,2\}^\mathbb{N}\times S_β$ and $K_β$ has a unique invariant measure of maximal entropy when $1<β\leqβ_*$, where $β_*\approx 1.4656$ is the root of $x^3-x^2-1=0$. We also show existence of a $K_β$-invariant probability measure, absolutely continuous with respect to $m_1\otimes m_2 \otimes λ_2$, where $m_1, m_2$ are product measures on $\{0,1\}^\mathbb{N},\{0,1,2\}^\mathbb{N}$, respectively, and $λ_2$ is the normalized Lebesgue measure on $S_β$. For $3/2<β\leq β^*$, where $β^*\approx 1.5437$ is the root of $x^3-2x^2+2x=2$, there are radial holes in $S_β$. In this case, $K_β$ is defined on $\{0,1\}^\mathbb{N}\times S_β$. We also show that it has a unique invariant measure of maximal entropy.