Quantitative mixing results and inner functions

Abstract

We study in this paper estimates on the size of the sets of points which are well approximated by orbits of other points under certain dynamical systems. We apply the results obtained to the particular case of the dynamical system generated by inner functions in the unit disk of the complex plane.

The dynamical Borel-Cantelli lemma and the waiting time problems

“…Moreover this problem is related to the distribution of return times (the property holds when the distribution of return times in small balls tends to be exponential, see [15], see also [7] for other general relations between waiting time and recurrence time distribution). While in the literature results on Borel-Cantelli and Waiting time are somewhat similar (and sometime used togheter, as in [14]), as far as we know, no explicit general relations about these two concepts are stated.…”

“…Indeed is proved (see e.g. [5], [20], [6], [10] [14], [22], [17]) that in many kind of (more or less) hyperbolic or "fast" mixing systems, various sequences of geometrically nice sets have the BC property. The kind of sets which are interesting to be considered in this kind of problems are usually decreasing sequences of balls with the same center (these are also called shrinking targets, this approach has relations with the theory of approximation speed, see [8], [19]) or cylinders.…”

A note on isoperimetric inequalities of Gromov hyperbolic manifolds and graphs

“…It is well-known that isoperimetric inequalities are of interest in applied and pure mathematics [22,48]. The Cheeger isoperimetric inequality is related with many conformal invariants in Riemannian manifolds and graphs, namely the exponent of convergence, the bottom of the spectrum of the Laplace-Beltrami operator, Poincaré-Sobolev inequalities, and the Hausdorff dimensions of the sets of both escaping and bounded geodesics in negatively curved surfaces [4,13], [18, p. 228], [23,[27][28][29][30][31]43,46], [52, p. 333]. Isoperimetric inequality is also closely related to Ancona's project on the space of positive harmonic functions of Gromov-hyperbolic graphs and manifolds [7][8][9].…”

Quantitative recurrence properties and homogeneous self-similar sets

“…This motivates many authors to investigate the so-called quantitative recurrence properties. For instance, the dynamical Borel-Cantelli lemma [7,9], the first return time [1], the shrinking target problems [10,12], and etc. Among them, Boshernitzan [6] first quantified the recurrence rate of generic orbits for general MMPSs.…”

“…For the purpose of comparison, when we require {T n x} n≥1 returns to the neighborhoods of a chosen point x 0 rather than the initial point x, the problem becomes the so-called shrinking target problem (STP). One can refer to [7,9] for the measure aspect of STP, and to [10,12,18] for the dimension aspect.…”