Quantitative Poincaré recurrence in continued fraction dynamical system

Abstract

Let T: X → X be a transformation. For any x ∈ [0, 1) and r > 0, the recurrence time T _r (x) of x under T in its r-neighborhood is defined as

$$\tau _r (x) = \inf \{ k \geqslant 1:d(T^k (x),x) < r\} .$$

For 0 ⩽ α ⩽ β ⩽ β, let E(α, β) be the set of points with prescribed recurrence time as follows

$$E\left( {\alpha ,\beta } \right) = \left\{ {x \in X:\mathop {\lim \inf }\limits_{r \to 0} \frac{{\log \tau _r \left( x \right)}} {{ - \log r}} = \alpha , \mathop {\lim \sup }\limits_{r \to 0} \frac{{\log \tau _r \left( x \right)}} {{ - \log r}} = \beta } \right\}.$$

In this note, we consider the Gauss transformation T on [0, 1), and determine the size of E(α, β) by showing that dim _H E(α, β) =1 no matter what α and β are. This can be compared with Feng and Wu’s result [Nonlinearity, 14 (2001), 81–85] on the symbolic space.