The Distal and Asymptotic Sets for Continued Fractions

Abstract

For continued fraction dynamical system [Formula: see text], we give a classification of the underlying space [Formula: see text] according to the orbit of a given point [Formula: see text]. The sizes of all classes are determined from the viewpoints of measure, Hausdorff dimension and topology. For instance, the Hausdorff dimension of the distal set of [Formula: see text] is one and the Hausdorff dimension of the asymptotic set is either zero or [Formula: see text] according to [Formula: see text] is rational or not.

Chaotic sets and Hausdorff dimension for Lüroth expansions

“…Our research provides new similarities between both sets of expansions in terms of topological dynamics and Hausdorff dimension. In particular, we establish a complete analogue with the results of W. Liu, B. Li and S. Wang in [19,20] on the distal, asymptotic and Li-Yorke pairs for the Gauss map, in the case of the Lüroth transformation.…”

“…Main results and structure of the paper. Lately, W. Liu, B. Li and S. Wang in [19,20] studied the chaotic properties of ([0, 1), G ). In the following statement, we summarise [19,Theorem 1.3,Corollary 1.4] and [20,Theorem 1.1,1.2].…”

“…Recall that Theorems A, B, and D are full analogues of the main results in [19], [20]. It is thus natural to pose the problems above for ([0, 1), G ), where G is the usual Gauss map.…”