American Institute of Mathematical Sciences

Advanced
July  2008, 2(3): 465-470. doi: 10.3934/jmd.2008.2.465

Almost-everywhere convergence and polynomials

Michael Boshernitzan 1, and  Máté Wierdl 2,

1. 

Department of Mathematics, Rice Unviersity, Houston, TX 77005, United States
2. 

Department of Mathematical Sciences, 373 Dunn Hall, University of Memphis, Memphis, TN 38152-3240, United States

Received  November 2007 Revised  March 2008 Published  April 2008

Denote by $\Gamma$ the set of pointwise good sequences: sequences of real numbers $(a_k)$ such that for any measure--preserving flow $(U_t)_{t\in\mathbb R}$ on a probability space and for any $f\in L^\infty$, the averages $\frac{1}{n} \sum_{k=1}^{n} f(U_{a_k}x) $ converge almost everywhere.
    We prove the following two results.
1. If $f: (0,\infty)\to\mathbb R$ is continuous and if $(f(ku+v))_{k\geq 1}\in\Gamma$ for all $u, v>0$, then $f$ is a polynomial on some subinterval $J\subset (0,\infty)$ of positive length.
2. If $f: [0,\infty)\to\mathbb R$ is real analytic and if $(f(ku))_{k\geq 1}\in\Gamma$ for all $u>0$, then $f$ is a polynomial on the whole domain $[0,\infty)$.
    These results can be viewed as converses of Bourgain's polynomial ergodic theorem which claims that every polynomial sequence lies in $\Gamma$.
Keywords: polynomials., Pointwise ergodic theorems along subsequences.
Mathematics Subject Classification: Primary: 37A10, Secondary: 37A3.
Citation: Michael Boshernitzan, Máté Wierdl. Almost-everywhere convergence and polynomials. Journal of Modern Dynamics, 2008, 2 (3) : 465-470. doi: 10.3934/jmd.2008.2.465
[1]

Elon Lindenstrauss. Pointwise theorems for amenable groups. Electronic Research Announcements, 1999, 5: 82-90.

[2]

Asaf Katz. On mixing and sparse ergodic theorems. Journal of Modern Dynamics, 2021, 17: 1-32. doi: 10.3934/jmd.2021001
[3]

Edson A. Coayla-Teran, Salah-Eldin A. Mohammed, Paulo Régis C. Ruffino. Hartman-Grobman theorems along hyperbolic stationary trajectories. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 281-292. doi: 10.3934/dcds.2007.17.281
[4]

Tanja Eisner, Jakub Konieczny. Automatic sequences as good weights for ergodic theorems. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4087-4115. doi: 10.3934/dcds.2018178
[5]

Yuri Kifer. Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2687-2716. doi: 10.3934/dcds.2018113
[6]

Zuohuan Zheng, Jing Xia, Zhiming Zheng. Necessary and sufficient conditions for semi-uniform ergodic theorems and their applications. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 409-417. doi: 10.3934/dcds.2006.14.409
[7]

Julia Brettschneider. On uniform convergence in ergodic theorems for a class of skew product transformations. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 873-891. doi: 10.3934/dcds.2011.29.873
[8]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363
[9]

Yves Derriennic. Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the "central limit theorem''. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 143-158. doi: 10.3934/dcds.2006.15.143
[10]

Oliver Jenkinson. Ergodic Optimization. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 197-224. doi: 10.3934/dcds.2006.15.197
[11]

Alexandre Alves, Mostafa Salarinoghabi. On the family of cubic parabolic polynomials. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021121
[12]

Valentina Casarino, Paolo Ciatti, Silvia Secco. Product structures and fractional integration along curves in the space. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 619-635. doi: 10.3934/dcdss.2013.6.619
[13]

Gregory A. Chechkin, Tatiana P. Chechkina, Ciro D’Apice, Umberto De Maio. Homogenization in domains randomly perforated along the boundary. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 713-730. doi: 10.3934/dcdsb.2009.12.713
[14]

Haixia Yu. Hilbert transforms along double variable fractional monomials. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1433-1446. doi: 10.3934/cpaa.2019069
[15]

Fangzhou Cai, Song Shao. Topological characteristic factors along cubes of minimal systems. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5301-5317. doi: 10.3934/dcds.2019216
[16]

Tassilo Küpper. Bifurcation revisited along footprints of Jürgen Scheurle. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1031-1041. doi: 10.3934/dcdss.2020061
[17]

Jayadev S. Athreya, Gregory A. Margulis. Values of random polynomials at integer points. Journal of Modern Dynamics, 2018, 12: 9-16. doi: 10.3934/jmd.2018002
[18]

Takao Komatsu, Bijan Kumar Patel, Claudio Pita-Ruiz. Several formulas for Bernoulli numbers and polynomials. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021006
[19]

Elisavet Konstantinou, Aristides Kontogeorgis. Some remarks on the construction of class polynomials. Advances in Mathematics of Communications, 2011, 5 (1) : 109-118. doi: 10.3934/amc.2011.5.109
[20]

Shrihari Sridharan, Atma Ram Tiwari. The dependence of Lyapunov exponents of polynomials on their coefficients. Journal of Computational Dynamics, 2019, 6 (1) : 95-109. doi: 10.3934/jcd.2019004

2019 Impact Factor: 0.465

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences