On shrinking targets for piecewise expanding interval maps

Abstract

Abstract. For a map T : [0, 1] → [0, 1] with an invariant measure µ, we study, for a µ-typical x, the set of points y such that the inequality |T n x − y| < rn is satisfied for infinitely many n. We give a formula for the Hausdorff dimension of this set, under the assumption that T is piecewise expanding and µ φ is a Gibbs measure. In some cases we also show that the set has a large intersection property.

Recurrence to Shrinking Targets on Typical Self-Affine Fractals

“…by EP/J018260/1. unit interval [22] and in the dynamical system of continued fraction expansions [20]. In all of these cases the dimension of the recurring set is given as a zero point of a generalized pressure functional.…”

Hausdorff dimension of random limsup sets

“…The Hausdorff dimension of sets of this kind was studied for particular types of dynamical systems, see for instance . For the dynamical systems studied in those papers, the problem of determining the Hausdorff dimension of the set is rather close to that of this paper.…”

A general mass transference principle

“…Within Diophantine Approximation, it is reasonable to expect that Theorem 1 will enable the establishment of further Hausdorff measure statements relating to approximation on manifolds (see [3,4] and the references therein for more on this problem). Within Dynamical Systems, it is also reasonable to expect that Theorem 1 will allow one to study a wider class of shrinking target problems, in particular when our target is allowed to have a more exotic structure (see [20,22,23] and the references therein for more on this problem). We hope to return to these topics in a later work.…”