A logarithmic law for continued fractions with sequentially restricted entries

Abstract: Non-stationary shift spaces are basic models of sequential dynamical system who were intensively studied in order to construct symbolic models for ergodic automorphism (Vershik) or in the context of the isomorphism problem of shift spaces (Krieger). Recently, the focus moved towards thermodynamic formalism and related questions. A fundamental tool of thermodynamic formalism, Ruelle's operator theorem, has no immediate generalization to the non-stationary setting as invariant functions intrinsically may not exist. However, it is possible to establish geometric ergodicity for a family of ratios of operators.

This approach has applications to a classical problem in metric number theory. For a sequence $(\alpha_n)$ converging to $\infty$, set \[X_\alpha := \left\{ x = \frac{1}{x_1 + \frac{1}{x_2 + \cdots} } : x_n \in \mathbb N, x_n \geq \alpha_n \hbox{ for all } n \right\}.\] That is, $X$ is the subset of $[0,1]$ such that the $n$-th entry of the continued fraction expansion of each element is bigger than or equal to $\alpha_n$. In this setting, for $\alpha_n \gg n^{1+\epsilon}$, the geometric ergodicity implies a law of the iterated logarithm for square integrable functions from geometric ergodicity. If, in addition, $(\alpha_n)$ does not behave too wildly, the reference measure is absolutely contiunous with respect to the Hausdorff measure (and the Hausdorff dimension is $1/2$). \\

dynamical systems

Audience: researchers in the topic

Dynamical systems seminar at the Jagiellonian University