Bounds for multiple recurrence rate and dimension. (English) Zbl 1459.37003

Hirayama, Michihiro

Tokyo J. Math. 42, No. 1, 239-253 (2019).

For a probability measure-preserving system \((X,\mathcal{B},\mu,T)\) and a set \(A\in\mathcal{B}\) with \(\mu(A)>0\), the Poincaré recurrence theorem states that \(\mu\)-almost every point in \(A\) returns to \(A\) infinitely often. If the space \(X\) has a compatible metric \(d\) for which \(\mathcal{B}\) is the Borel \(\sigma\)-algebra then a result of M. D. Boshernitzan [Invent. Math. 113, No. 3, 617–631 (1993; Zbl 0839.28008)] gives qualitative information about the closeness of returns, showing that \(\liminf_{n\to\infty}d(x,T^nx)=0\) for \(\mu\)-almost every \(x\in X\) and, under the geometric hypothesis that the \(\alpha\)-Hausdorff measure of \(X\) is finite for some \(\alpha>0\), goes on to show that \(\liminf_{n\to\infty}\bigl(n^{1/\alpha}d(x,T^nx)\bigr)<\infty\) for \(\mu\)-almost every \(x\in X\). L. Barreira and B. Saussol [Commun. Math. Phys. 219, No. 2, 443–463 (2001; Zbl 1007.37012)] gave estimates for the first return time to a metric ball, and D. H. Kim [Nonlinearity 22, No. 1, 1–9 (2009; Zbl 1167.37006)] generalized Boshernitzan’s results to actions of countable discrete groups.
Here a multiple and simultaneous analogue of these results are found; the results are too complicated to state here but the flavor is to find quantitative versions of statements of the form \(\liminf_{n\to\infty}\mathrm{diam} \{x,T^nx,T^{2n}x,\dots,T^{Ln}x\}=0\) for \(\mu\)-almost every \(x\in X\). The methods used are diverse and they include results of W. T. Gowers [Geom. Funct. Anal. 11, No. 3, 465–588 (2001); Erratum 11, No. 4, 869 (2001; Zbl 1028.11005)] on the quantitative Szemerédi theorem to study long simultaneous return.

Reviewer: Thomas B. Ward (Leeds)

MSC:

37A05	Dynamical aspects of measure-preserving transformations
37A10	Dynamical systems involving one-parameter continuous families of measure-preserving transformations
37A30	Ergodic theorems, spectral theory, Markov operators
37C35	Orbit growth in dynamical systems
37C45	Dimension theory of smooth dynamical systems
37B20	Notions of recurrence and recurrent behavior in topological dynamical systems

Keywords:

multiple recurrence; first simultaneous return time; Hausdorff dimension

Citations:

Zbl 0839.28008; Zbl 1007.37012; Zbl 1167.37006; Zbl 1028.11005

References: