Diophantine approximation by orbits of expanding Markov maps

Abstract

Abstract. Given a dynamical system ([0, 1], T ), the distribution properties of the orbits of real numbers x ∈ [0, 1] under T constitute a longstanding problem. In 1995, Hill and Velani introduced the "shrinking targets" theory, which aims at investigating precisely the Hausdorff dimensions of sets whose orbits are close to some fixed point. In this paper, we study the sets of points well-approximated by orbits {T n x} n≥0 , where T is an expanding Markov map with finite partitions supported by the whole interval [0,1]. The values of the dimensions of sets of well-approximable points are described using the multifractal properties of Gibbs measures invariant under the action of T . This study can be viewed as a moving shrinking targets problem.

Metric Diophantine approximation on the middle-third Cantor set

“…We refer to Theorem 8 for details. If the real numbers ξ are not restricted to belong to the Cantor set, then one recovers a much easier situation already studied by various authors including Bugeaud [7], Schmeling and Troubetzkoy [30], Fan, Schmeling, and Troubetzkoy [17], and also by Liao and Seuret [26]. Most of the proofs are postponed to Sections 7 and 8, while Section 6 is devoted to concluding observations and a brief discussion on further problems.…”

On shrinking targets for piecewise expanding interval maps

“…This means that the set E(x, r) belongs for some 0 < s < 1 to the class G s of G δ -sets, with the property that any countable intersection of bi-Lipschitz images of sets in G s has Hausdorff dimension at least s. See Falconer's paper [3] for more details about those classes of sets. The large intersection property was not proved in any of the papers [4], [6] and [7].…”

Hausdorff dimension of affine random covering sets in torus

“…In general metric spaces, the random coverings by balls have been studied by Hoffman-Jörgensen [15]. Recent contributions to the topic include various types of dynamical models, see Fan, Schmeling and Troubetzkoy [11], Jonasson and Steif [17] and Liao and Seuret [23].…”