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The natural extension of the random beta-transformation
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 43 / Issue 11 / November 2023
- Published online by Cambridge University Press:
- 04 November 2022, pp. 3861-3896
- Print publication:
- November 2023
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We construct a geometrico-symbolic version of the natural extension of the random
$\beta $-transformation introduced by Dajani and Kraaikamp [Random 
$\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 $-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 $-automorphisms are Bernoulli shifts. Acta Math. Acad. Sci. Hungar. 24 (1973), 273–278] about the greedy transformation.
SIMULTANEOUS DYNAMICAL DIOPHANTINE APPROXIMATION IN BETA EXPANSIONS
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 102 / Issue 2 / October 2020
- Published online by Cambridge University Press:
- 08 January 2020, pp. 186-195
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- October 2020
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Let
$\unicode[STIX]{x1D6FD}>1$ be a real number and define the 
$\unicode[STIX]{x1D6FD}$-transformation on 
$[0,1]$ by 
$T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$. Let 
$f:[0,1]\rightarrow [0,1]$ and 
$g:[0,1]\rightarrow [0,1]$ be two Lipschitz functions. The main result of the paper is the determination of the Hausdorff dimension of the set where
$$\begin{eqnarray}W(f,g,\unicode[STIX]{x1D70F}_{1},\unicode[STIX]{x1D70F}_{2})=\big\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-f(x)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{1}(x)},|T_{\unicode[STIX]{x1D6FD}}^{n}y-g(y)|<\unicode[STIX]{x1D6FD}^{-n\unicode[STIX]{x1D70F}_{2}(y)}~\text{for infinitely many}~n\in \mathbb{N}\big\},\end{eqnarray}$$
$\unicode[STIX]{x1D70F}_{1}$, 
$\unicode[STIX]{x1D70F}_{2}$ are two positive continuous functions with 
$\unicode[STIX]{x1D70F}_{1}(x)\leq \unicode[STIX]{x1D70F}_{2}(y)$ for all 
$x,y\in [0,1]$.
