2021/01/06 by Kalle, Charlene, Maggioni, Marta · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2101.01982
We introduce a family of random c-Lüroth transformations \Lc\c ∈ [0, \frac12], 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 Lc produces for each c ≤ \frac25 and for Lebesgue almost all x ∈ [c,1] uncountably many different generalised Lüroth expansions that can be investigated simultaneously. Moreover, for c= \frac1ℓ, for ℓ ∈ ℕ≥ 3 ∪ \∞\, Lebesgue almost all x have uncountably many universal generalised Lüroth expansions with digits less than or equal to ℓ. For c=0 we show that typically the speed of convergence to an irrational number x, of the sequence of Lüroth approximants generated by L0, 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 ∈ \mathbb Q the map Lc 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.