No abstract

For beta-transformations, we prove that the Lebesgue measure of any measurable scrambled set is zero, and there exists a scrambled set with full Hausdorff dimension.

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.

No abstract

For any [Formula: see text], let [Formula: see text] be the [Formula: see text]-transformation dynamical system. For any sequence [Formula: see text], we investigate the following badly approximable set: [Formula: see text] In this paper, we determine the Lebesgue measure and Hausdorff dimension of the set [Formula: see text] completely for any [Formula: see text] and any sequence [Formula: see text].