On approximation by random Lüroth expansions

Abstract

We introduce a family of random $c$-Lüroth transformations $\{L_c{\}}_{c\in [0,\frac{1}{2}]}$, obtained by randomly combining the standard and alternating Lüroth maps with probabilities $p$ and $1-p$, $0<p<1$, both defined on the interval $[c,1]$. We prove that the pseudo-skew product map $L_c$ produces for each $c\le \frac{2}{5}$ and for Lebesgue almost all $x\in [c,1]$ uncountably many different generalised Lüroth expansions that can be investigated simultaneously. Moreover, for $c=\frac{1}{\ell }$, for $\ell \in {\mathbb{N}}_{\ge 3}\cup \{\infty \}$, Lebesgue almost all $x$ have uncountably many universal generalised Lüroth expansions with digits less than or equal to $\ell$. For $c=0$ we show that typically the speed of convergence to an irrational number $x$, of the sequence of Lüroth approximants generated by $L_0$, is equal to that of the standard Lüroth approximants; and that the quality of the approximation coefficients depends on $p$ and varies continuously between the values for the alternating and the standard Lüroth map. Furthermore, we show that for each $c\in \mathbb{Q}$ the map $L_c$ admits a Markov partition. For specific values of $c>0$, we compute the density of the stationary measure and we use it to study the typical speed of convergence of the approximants and the digit frequencies.