We construct a geometrico-symbolic version of the natural extension of the random \beta $\beta$-transformation introduced by Dajani and Kraaikamp [Random \beta $\beta$-expansions. Ergod. Th. & Dynam. Sys. 23(2) (2003) 461–479]. This construction provides a new proof of the existence of a unique absolutely continuous invariant probability measure for the random \beta $\beta$-transformation, and an expression for its density. We then prove that this natural extension is a Bernoulli automorphism, generalizing to the random case the result of Smorodinsky [\beta $\beta$-automorphisms are Bernoulli shifts. Acta Math. Acad. Sci. Hungar. 24 (1973), 273–278] about the greedy transformation.

Let \unicode[STIX]{x1D6FD}>1$\unicode[STIX]{x1D6FD}>1$ be a real number and define the \unicode[STIX]{x1D6FD}$\unicode[STIX]{x1D6FD}$-transformation on [0,1]$[0,1]$ by T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$. Let f:[0,1]\rightarrow [0,1]$f:[0,1]\rightarrow [0,1]$ and g:[0,1]\rightarrow [0,1]$g:[0,1]\rightarrow [0,1]$ be two Lipschitz functions. The main result of the paper is the determination of the Hausdorff dimension of the set

\begin{eqnarray}W(f,g,\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})=\big\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-f(x)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{1}(x)},|T_{\unicode[STIX]{x1D6FD}}^{n}y-g(y)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{2}(y)}~\text{for infinitely many}~n\in \mathbb{N}\big\},\end{eqnarray} $$\begin{eqnarray}W(f,g,\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})=\big\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-f(x)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{1}(x)},|T_{\unicode[STIX]{x1D6FD}}^{n}y-g(y)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{2}(y)}~\text{for infinitely many}~n\in \mathbb{N}\big\},\end{eqnarray}$$

\unicode[STIX]{x1D70F}_{1}$\unicode[STIX]{x1D70F}_{1}$

\unicode[STIX]{x1D70F}_{2}$\unicode[STIX]{x1D70F}_{2}$

\unicode[STIX]{x1D70F}_{1}(x)\leq \unicode[STIX]{x1D70F}_{2}(y)$\unicode[STIX]{x1D70F}_{1}(x)\leq \unicode[STIX]{x1D70F}_{2}(y)$

x,y\in [0,1]$x,y\in [0,1]$