No Access

Diffusive behavior of ergodic sums over rotations

Jean-Pierre Conze

CNRS, IRMAR – UMR 6625, University of Rennes, F-35000 Rennes, France

E-mail Address: conze@univ-rennes1.fr

Search for more papers by this author

Stefano Isola

Scuola di Scienze e Tecnologie, Universitá di Camerino, via Madonna delle Carceri, 62032 Camerino, Italy

E-mail Address: stefano.isola@unicam.it

Search for more papers by this author

, and

Stéphane Le Borgne

CNRS, IRMAR – UMR 6625, University of Rennes, F-35000 Rennes, France

E-mail Address: stephane.leborgne@univ-rennes1.fr

Search for more papers by this author

https://doi.org/10.1142/S0219493719500163Cited by:3 (Source: Crossref)

Abstract

For a rotation by an irrational $α$ $α$ on the circle and a BV function $φ$ $φ$ , we study the variance of the ergodic sums $S_{L} φ (x) : = \sum_{j = 0}^{L - 1} φ (x + j α)$ $S_{L} φ (x) : = \sum_{j = 0}^{L - 1} φ (x + j α)$ . When $α$ $α$ is not of constant type, we construct sequences $(L_{N})$ $(L_{N})$ such that, at some scale, the ergodic sums $S_{L_{N}} φ$ $S_{L_{N}} φ$ satisfy an ASIP. Explicit non-degenerate examples are given with an application to the rectangular periodic billiard in the plane.

Dedicated to the memory of Eugene Gutkin

Keywords:

AMSC: 11A55, 42A55, 60F05, 60F17, 37D50

References

1. C. Aistleitner and I. Berkes , On the central limit theorem for $f (n_{k} x)$ $f (n_{k} x)$ , Probab. Theory Related Fields 146 (2010) 267–289. Web of Science, Google Scholar
2. I. Berkes , On the asymptotic behaviour of $\sum f (n_{k} x)$ $\sum f (n_{k} x)$ : I. Main theorems, II. Applications, Z. Wahrscheinlichkeitstheorie verw. Gebiete 34 (1976) 319–345, 347–365. Web of Science, Google Scholar
3. I. Berkes and W. Philipp , An a.e. invariance principle for lacunary series $f (n_{k} x)$ $f (n_{k} x)$ , Acta Math. Acad. Sci. Hungar. 34 (1979) 141–155. Web of Science, Google Scholar
4. J.-P. Conze and E. Gutkin , On recurrence and ergodicity for geodesic flows on non-compact periodic polygonal surfaces, Ergodic Theory Dynam. Syst. 32(2) (2012) 491–515. Web of Science, Google Scholar
5. J.-P. Conze and S. Le Borgne , Limit law for some modified ergodic sums, Stoch. Dynam. 11 (2011) 107–133. Link, Web of Science, Google Scholar
6. J.-P. Conze and S. Le Borgne, On the central limit theorem for rotations and BV functions, preprint (2018), arXiv:1804.09929. Google Scholar
7. A. Denjoy , Les trajectoires à la surface du tore, C. R. Acad. Sci. Paris 223 (1946) 5–8. Google Scholar
8. V. F. Gaposhkin , On the central limit theorem for some weakly dependent sequences, Teor. Verojatn. i Primenen 15 (1970) 666–684 (in Russian). Google Scholar
9. S. Gouëzel , Almost sure invariance principle for dynamical systems by spectral methods, Ann. Probab. 38 (2010) 1639–1671. Web of Science, Google Scholar
10. J. Hardy and J. Weber , Diffusion in a periodic wind-tree model, J. Math. Phys. 21 (1980) 1802–1808. Web of Science, Google Scholar
11. F. Huveneers , Subdiffusive behavior generated by irrational rotations, Ergodic Theory Dynam. Syst. 29 (2009) 1217–1233. Web of Science, Google Scholar
12. S. Isola , Dispersion properties of ergodic translations, Int. J. Math. Math. Sci. Article ID: 20568 (2006), 20 pp. Google Scholar
13. M. Kac, R. Salem and A. Zygmund , A gap theorem, Trans. Amer. Math. Soc. 63 (1948) 235–243. Web of Science, Google Scholar
14. A. Ya. Khinchin , Continued Fractions (Dover Publications, Mineola, N.Y., 1997). Google Scholar
15. W. Philipp and W. F. Stout , Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc. 2 (1975). Web of Science, Google Scholar
16. R. Salem and A. Zygmund , On lacunary trigonometric series. II, Proc. Natl. Acad. Sci. U.S.A. 34 (1948) 54–62. Web of Science, Google Scholar