Almost-everywhere convergence and polynomials

Previous Article
Hausdorff dimension for ergodic measures of interval exchange transformations
JMD Home
This Issue
Next Article
Displacement energy of coisotropic submanifolds and Hofer's geometry

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

Abstract
Related Papers

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]	Tanja Eisner, Jakub Konieczny. Automatic sequences as good weights for ergodic theorems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4087-4115. doi: 10.3934/dcds.2018178
[3]	Edson A. Coayla-Teran, Salah-Eldin A. Mohammed, Paulo Régis C. Ruffino. Hartman-Grobman theorems along hyperbolic stationary trajectories. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 281-292. doi: 10.3934/dcds.2007.17.281
[4]	Yuri Kifer. Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2687-2716. doi: 10.3934/dcds.2018113
[5]	Zuohuan Zheng, Jing Xia, Zhiming Zheng. Necessary and sufficient conditions for semi-uniform ergodic theorems and their applications. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 409-417. doi: 10.3934/dcds.2006.14.409
[6]	Julia Brettschneider. On uniform convergence in ergodic theorems for a class of skew product transformations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 873-891. doi: 10.3934/dcds.2011.29.873
[7]	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 - A, 2006, 15 (1) : 143-158. doi: 10.3934/dcds.2006.15.143
[8]	Oliver Jenkinson. Ergodic Optimization. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 197-224. doi: 10.3934/dcds.2006.15.197
[9]	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
[10]	Nathaniel D. Emerson. Dynamics of polynomials with disconnected Julia sets. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 801-834. doi: 10.3934/dcds.2003.9.801
[11]	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
[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]	Tassilo Küpper. Bifurcation revisited along footprints of Jürgen Scheurle. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-11. doi: 10.3934/dcdss.2020061
[16]	Fangzhou Cai, Song Shao. Topological characteristic factors along cubes of minimal systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5301-5317. doi: 10.3934/dcds.2019216
[17]	Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757
[18]	Roy Adler, Bruce Kitchens, Michael Shub. Stably ergodic skew products. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 349-350. doi: 10.3934/dcds.1996.2.349
[19]	Alexandre I. Danilenko, Mariusz Lemańczyk. Spectral multiplicities for ergodic flows. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4271-4289. doi: 10.3934/dcds.2013.33.4271
[20]	Doǧan Çömez. The modulated ergodic Hilbert transform. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 325-336. doi: 10.3934/dcdss.2009.2.325

2017 Impact Factor: 0.425

American Institute of Mathematical Sciences