×

Some results on nonlinear recurrence. (English) Zbl 0803.28011

The authors are interested in continuous time analogies of the nice properties that discrete measure-preserving systems exhibit along polynomial sequences. (For example, if \(p(t)\) is a polynomial with integer coefficients and \(f\in L^ 2(X,{\mathcal B},\mu)\) then the averages \({1\over N} \sum^ N_{n=1} f(T^{p(n)} x)\) converge in \(L^ 2\)-norm, where \(T\) is a measure-preserving transformation.) The situation turns out to be quite different for \(\mathbb{R}\) actions.
Let \(\{S^ t\}_{t\in \mathbb{R}}\) be a continuous one-parameter flow of measure-preserving transformations of a Lebesgue probability space \((X,{\mathcal B},\mu)\). A sequence of real numbers \(\{\lambda_ n\}\) is said to be \(L^ p(T)\)-ergodic \((\{\lambda_ n\}\in E_ p(T))\), \(1\leq p\leq \infty\), if for every \(f\in L^ p(T)\), \(T= \mathbb{R}/\mathbb{Z}\), the sequence \(A_ N(f)= {1\over N} \sum^ N_{n=1} f(x+ \lambda_ n)\) is convergent a.e. (where \(x+ \lambda_ n\) is taken modulo 1).
Theorem A. Let \(\lambda= \{\lambda_ n\}\) be a sequence of real numbers independent over \(\mathbb{Q}\). Then \(\lambda\not\in E_ \infty(T)\).
Theorem B. For any \(\alpha\in \mathbb{Q}/\mathbb{Z}\), we have \(n^ \alpha= \{n^ \alpha\}\not\in E_ \infty(T)\).
Theorem C. Let \(\alpha_ 1,\alpha_ 2,\dots,\alpha_ k\in (0,1)\), \(\alpha_ i\neq \alpha_ j\) for \(i\neq j\). Let \(\{S^ t\}\) be an ergodic measure-preserving continuous flow acting on \((X,{\mathcal B},\mu\}\). Then for any \(f_ 1,f_ 2,\dots,f_ k\in L^ \infty(X,{\mathcal B},\mu)\) one has \[ \lim_{N\to \infty} \left\|{1\over N} \sum^ N_{n=1} \prod^ k_{i=1} f_ i(S^{n^{\alpha_ i}}x)- \prod^ k_{i=1} \int_ X f_ i d\mu\right\|_{L^ 2}= 0. \] The authors also obtain interesting consequences of these results and results on densities paralleling the discrete case.

MSC:

28D10 One-parameter continuous families of measure-preserving transformations
28D05 Measure-preserving transformations
37A99 Ergodic theory
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] [BE1] V. Bergelson,Weakly mixing PET, Ergodic Theory & Dynamical Systems7 (1987), 337–349.
[2] [BE2] V. Bergelson,Independence properties of continuous flows, inAlmost Everywhere Convergence, Academic Press, New York, 1989, pp. 121–130.
[3] [BE3] V. Bergelson,Ergodic Ramsey theory, Contemporary Math.75 (1987), 63–87.
[4] [BOU1] J. Bourgain,On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math.61 (1988), 39–72. · Zbl 0642.28010
[5] [BOU2] J. Bourgain,On the pointwise ergodic theorem on L p for arithmetic sets, Israel J. Math.61 (1988), 73–84. · Zbl 0642.28011
[6] [BOU3] J. Bourgain,Almost sure convergence and bounded entropy, Israel J. Math.73 (1988), 79–97. · Zbl 0677.60042
[7] [C] P. Csillag,Über die gleichmässige Verteilung nichtganzer positiver Potenzen mod 1, Acta Litt. Sci. Szeged5 (1930), 13–18. · JFM 56.0898.04
[8] [CFS] I. Cornfeld, S. Fomin and Ya. Sinai,Ergodic Theory, Springer-Verlag, New York, 1981.
[9] [F] H. Furstenberg,Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981. · Zbl 0459.28023
[10] [J] R. Jones,Inequalities for pairs of ergodic transformations, Radovi Mathematički4 (1988), 55–61. · Zbl 0642.28009
[11] [P] K. Petersen,The ergodic theorem with time compression, J. Analyse Math.51 (1988), 228–244;erratum: J. Analyse Math.55 (1990), 297. · Zbl 0662.28011
[12] [R] I. Richards,An application of Galois theory to elementary arithmetics, Adv. in Math.13 (1974), 268–273. · Zbl 0292.12002
[13] [W] H. Weyl,Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann.77 (1916), 313–352. · JFM 46.0278.06
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.