Illinois Journal of Mathematics

Multiple ergodic averages for flows and an application

Amanda Potts

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Abstract

We show the $L^2$-convergence of continuous time ergodic averages of a product of functions evaluated at return times along polynomials. These averages are the continuous time version of the averages appearing in Furstenberg's proof of Szemerédi’s Theorem. For each average, we show that it is sufficient to prove convergence on special factors, the Host-Kra factors, which have the structure of a nilmanifold. We also give a description of the limit. In particular, if the polynomials are independent over the real numbers then the limit is the product of the integrals. We further show that if the collection of polynomials has “low complexity”, then for every set $E$ of real numbers with positive density and for every $\delta>0$, the set of polynomial return times for the “$\delta$-thickened” set $E_{\delta}$ has bounded gaps. We give bounds for the flow average complexity and show that in some cases the flow average complexity is strictly less than the discrete average complexity.

Article information

Source
Illinois J. Math., Volume 55, Number 2 (2011), 589-621.

Dates
First available in Project Euclid: 1 February 2013

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1359762404

Digital Object Identifier
doi:10.1215/ijm/1359762404

Mathematical Reviews number (MathSciNet)
MR2941317

Zentralblatt MATH identifier
1300.37003

Subjects
Primary: 37A10: One-parameter continuous families of measure-preserving transformations
Secondary: 11B05: Density, gaps, topology 05D10: Ramsey theory [See also 05C55]

Citation

Potts, Amanda. Multiple ergodic averages for flows and an application. Illinois J. Math. 55 (2011), no. 2, 589--621. doi:10.1215/ijm/1359762404. https://projecteuclid.org/euclid.ijm/1359762404


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References

  • W. Ambrose and S. Kakutani, Structure and continuity of measurable flows, Duke Math. J. 9 (1942), 25–42.
  • L. Auslander, L. Green and F. Hahn, Flows on homogeneous spaces, Annals of Mathematics Studies, vol. 53, Princeton University Press, Princeton, NJ, 1963. With the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg.
  • V. Bergelson, Weakly mixing PET, Ergodic Theory Dynam. Systems 7 (1987), 337–349.
  • V. Bergelson, B. Host and B. Kra, Multiple recurrence and nilsequences, Invent. Math. 160 (2005), 261–303. With an appendix by Imre Ruzsa.
  • V. Bergelson, A. Leibman and E. Lesigne, Complexities of finite families of polynomials, Weyl systems, and constructions in combinatorial number theory, J. Anal. Math. 103 (2007), 47–92.
  • J. Bourgain, A Szemerédi type theorem for sets of positive density in $\mathbf{R}\sp k$, Israel J. Math. 54 (1986), 307–316.
  • J. Conze and E. Lesigne, Théorèmes ergodiques pour des mesures diagonales, Bull. Soc. Math. France 112 (1984), 143–175.
  • N. Frantzikinakis, Multiple ergodic averages for three polynomials and applications, Trans. Amer. Math. Soc. 360 (2008), 5435–5475.
  • N. Frantzikinakis and B. Kra, Convergence of multiple ergodic averages for some commuting transformations, Ergodic Theory Dynam. Systems 25 (2005), 799–809.
  • N. Frantzikinakis and B. Kra, Polynomial averages converge to the product of integrals, Israel J. Math. 148 (2005), 267–276.
  • H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256.
  • H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, NJ, 1981.
  • H. Furstenberg, Y. Katznelson and B. Weiss, Ergodic theory and configurations in sets of positive density, Mathematics of Ramsey theory, Algorithms Combin., vol. 5, Springer, Berlin, 1990, pp. 184–198.
  • H. Furstenberg and B. Weiss, A mean ergodic theorem for $(1/N)\sum\sp N\sb{n=1}f(T\sp nx)g(T\sp{n\sp2}x)$, Convergence in ergodic theory and probability (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., vol. 5, de Gruyter, Berlin, 1996, pp. 193–227.
  • S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978.
  • B. Host and B. Kra, Convergence of polynomial ergodic averages, Israel J. Math. 149 (2005), 1–19.
  • B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2) 161 (2005), 397–488.
  • A. Leibman, Convergence of multiple ergodic averages along polynomials of several variables, Israel J. Math. 146 (2005), 303–315.
  • A. Leibman, Pointwise convergence of ergodic averages for polynomial actions of ${\Bbb Z}\sp d$ by translations on a nilmanifold, Ergodic Theory Dynam. Systems 25 (2005), 215–225.
  • A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems 25 (2005), 201–213.
  • A. Leibman, Orbit of the diagonal in the power of a nilmanifold, Trans. Amer. Math. Soc. 362 (2010), 1619–1658.
  • A. I. Malcev, On a class of homogeneous spaces, Amer. Math. Soc. Translation 1951 (1951), 33.
  • W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math. 91 (1969), 757–771.
  • C. Pugh and M. Shub, Ergodic elements of ergodic actions, Compositio Math. 23 (1971), 115–122.
  • M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), 235–280.
  • N. A. Shah, Limit distributions of polynomial trajectories on homogeneous spaces, Duke Math. J. 75 (1994), 711–732.
  • J. G. van der Corput, Diophantische Ungleichungen. I. Zur Gleichverteilung Modulo Eins, Acta Math. 56 (1931), 373–456.
  • H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313–352.
  • T. Ziegler, A non-conventional ergodic theorem for a nilsystem, Ergodic Theory Dynam. Systems 25 (2005), 1357–1370.
  • T. Ziegler, Nilfactors of $\Bbb R\sp m$-actions and configurations in sets of positive upper density in $\Bbb R\sp m$, J. Anal. Math. 99 (2006), 249–266.