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**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.

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 |

Full Text:
Euclid

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