Exponent of Convergence of a Sequence of Ergodic Averages

I. V. Podvigin

Matematičeskie zametki, Tome 112 (2022) no. 2, pp. 251-262.

Résumé

For a sequence of ergodic averages, we consider its exponent of convergence, which is a numerical characteristic of two-sided power-law estimates of the rate of pointwise convergence of this sequence. Criteria for the boundary values 1 and $\infty$ of the exponent of convergence are given. Functions cohomologous to zero with a given the exponent of convergence are also described.

Détail

Keywords: Birkhoff's ergodic theorem, rates of convergence in ergodic theorems, the exponent of convergence, Tanny–Woś spaces.

Bibliographie

[1] A. G. Kachurovskii, I. V. Podvigin, “Ob izmerenii skorostei skhodimosti v ergodicheskoi teoreme Birkgofa”, Matem. zametki, 106:1 (2019), 40–52 | DOI | MR | Zbl

[2] I. V. Podvigin, “Lower bound of the supremum of ergodic averages for $ \mathbb{Z}^d$ and $\mathbb{R}^d$-actions”, Sib. elektron. matem. izv., 17 (2020), 626–636 | DOI | Zbl

[3] A. G. Kachurovskii, I. V. Podvigin, A. A. Svischev, “Zakon nulya ili edinitsy dlya skorostei skhodimosti v ergodicheskoi teoreme Birkgofa s nepreryvnym vremenem”, Matem. tr., 24:2 (2021), 65–80 | DOI

[4] G. Polia, G. Sege, Izoperimetricheskie neravenstva v matematicheskoi fizike, Fizmatlit, M., 1962 | MR | Zbl

[5] B. Ya. Levin, Raspredelenie kornei tselykh funktsii, Gostekhizdat, M., 1956 | MR | Zbl

[6] V. F. Gaposhkin, “O zavisimosti skorosti skhodimosti v usilennom zakone bolshikh chisel dlya statsionarnykh protsessov ot skorosti ubyvaniya korrelyatsionnoi funktsii”, Teoriya veroyatn. i ee primen., 26:4 (1981), 720–733 | MR | Zbl

[7] A. G. Kachurovskii, “Skorosti skhodimosti v ergodicheskikh teoremakh”, UMN, 51:4 (1996), 73–124 | DOI | MR | Zbl

[8] R. Rühr, R. Shi, “Quantitative multiple pointwise convergence and effective multiple correlations”, J. Differential Equations, 285 (2021), 1–16 | DOI | MR | Zbl

[9] M. Lin, R. Sine, “Ergodic theory and the functional equation $(I-T)x=y$”, J. Operator Theory, 10 (1983), 153–166 | MR | Zbl

[10] J. Woś, “The filling scheme and the ergodic theorems of Kesten and Tanny”, Colloq. Math., 52 (1987), 263–276 | DOI | MR | Zbl

[11] A. G. Kachurovskii, I. V. Podvigin, A. A. Svischev, “Maksimalnaya potochechnaya skorost skhodimosti v ergodicheskoi teoreme Birkgofa”, Teoriya predstavlenii, dinamicheskie sistemy, kombinatornye metody. XXXI, Zap. nauchn. sem. POMI, 498, POMI, SPb., 2020, 18–25 | MR

[12] A. B. Katok, Combinatorial Constructions in Ergodic Theory and Dynamics, Amer. Math. Soc., Providence, RI, 2003 | MR | Zbl

[13] Y. Derriennic, M. Lin, “Fractional Poisson equations and ergodic theorems for fractional coboundaries”, Israel J. Math., 123 (2001), 93–130 | DOI | MR | Zbl

[14] L. Keipers, G. Niderreiter, Ravnomernoe raspredelenie posledovatelnostei, Mir, M., 1985 | MR | Zbl

[15] P. Liardet, D. Volny, “Sums of continuous and differentiable functions in dynamical systems”, Israel J. Math., 98 (1997), 29–60 | DOI | MR | Zbl

[16] D. Tanny, “A zero-one law for stationary sequences”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 30 (1974), 139–148 | DOI | MR | Zbl

[17] Y. Derriennic, “Ergodic theorem, reversibility and the filling scheme”, Colloq. Math., 118:2 (2010), 599–608 | DOI | MR | Zbl

[18] J. Wolfowitz, “The moments of recurrence time”, Proc. Amer. Math. Soc., 18 (1967), 613–614 | DOI | MR | Zbl

[19] T. Adams, J. Rosenblatt, “Joint coboundaries”, Dynamical Systems, Ergodic Theory, and Probability: In Memory of Kolya Chernov, Contemp. Math., 698, Amer. Math. Soc., Providence, RI, 2017, 5–33 | DOI | MR | Zbl

[20] T. Adams, J. Rosenblatt, Existence and Non-existence of Solutions to the Coboundary Equation for Measure Preserving Systems, 2019, arXiv: 1902.09045

[21] A. del Junco, J. Rosenblatt, “Counterexamples in ergodic theory and number theory”, Math. Ann., 245 (1979), 185–197 | DOI | MR | Zbl