A Dichotomy Law for the Diophantine Properties in ‐dynamical Systems

Abstract

Let β > 1 be a real number and define the β-transformation on [0, 1] by T β : x → βx mod 1. Further, definefor infinitely many n} andfor infinitely many n}, where Ψ : N → R >0 is a positive function such that Ψ(n) → 0 as n → ∞. In this paper, we show that each of the above sets obeys a Jarník-type dichotomy, that is, the generalised Hausdorff measure is either zero or full depending upon the convergence or divergence of a certain series. This work completes the metrical theory of these sets.

Simultaneous Dynamical Diophantine Approximation in Beta Expansions

“…It is easy to see that the set W(T, ψ) can also be viewed as the collection of points in X whose T -orbit hits a shrinking target infinitely often. For background and more results on shrinking target problems, the reader is referred to [3,4,6,7,[12][13][14][15].…”

Two-Dimensional Shrinking Target Problem in Beta-Dynamical Systems

“…Following the work of Hill and Velani [8,9], the Hausdorff dimension of the set D(T, ϕ) has been determined for many dynamical systems, from the system of rational expanding maps on their Julia sets to conformal iterated function systems [19]. We refer the reader to [5] for a comprehensive discussion regarding the Hausdorff dimension of various dynamical systems. In this paper, we confine ourself to the two dimensional shrinking target problem in the beta dynamical system with a general error of approximation.…”

Inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems

“…satisfies (1.5) and this set has Hausdorff dimension 1 + 1 1+α with α given in (1.6). Coons, Hussain and Wang [4] extended this result to the generalised Hausdorff measure.…”