Induced random $\beta $-transformation

Simon Baker¹ ; Karma Dajani²

¹ Department of Mathematics and Statistics University of Reading Reading, RG6 6AX, UK
² Department of Mathematics Utrecht University 3508 TA Utrecht, The Netherlands

Acta Arithmetica, Tome 178 (2017) no. 1, pp. 1-14

Résumé

We study the first return map defined on the switch region induced by the greedy map and the lazy map. In particular we study the allowable sequences of return times, and when the first return map is a generalised Lüroth series transformation. We show that there exists a countable collection $(\mathcal{I}_{n})_{n=1}^{\infty}$ of disjoint intervals such that all sequences of return times are permissible if and only if $\beta\in \mathcal{I}_{n}$ for some $n$. Moreover, we show that there exists a set $M\subseteq(1,2)$ of Hausdorff dimension $1$ and Lebesgue measure zero for which the first return map is a generalised Lüroth series transformation if and only if $\beta\in M$.

Détail

DOI : 10.4064/aa8306-8-2016

Keywords: study first return map defined switch region induced greedy map lazy map particular study allowable sequences return times first return map generalised roth series transformation there exists countable collection mathcal infty disjoint intervals sequences return times permissible only beta mathcal moreover there exists set subseteq hausdorff dimension lebesgue measure zero which first return map generalised roth series transformation only beta