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

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.


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
Full Text: Euclid


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