Multiplicative Diophantine approximation in the parameter space of beta-dynamical system

Abstract: Beta-transformation is a special kind of expanding dynamics, the total information of which can be determined by the orbits of some critical points (e.g., the point 1). Letting $T_{\beta}$ be the beta-transformation with $\beta>1$ and $x$ be a fixed point in $(0,1]$, we consider the set of parameters $(\alpha, \beta)$, such that the multiple $\|T^n_{\alpha}(x)\|\|T^n_{\beta}(x)\|$ is well approximated or badly approximated. The Gallagher-type question, Jarník-type question as well as the badly approximable pairs, i.e., Littlewood-type question are studied in detail.

dynamical systemsnumber theory

Audience: researchers in the topic

( slides )

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