-
Previous Article
Displacement energy of coisotropic submanifolds and Hofer's geometry
- JMD Home
- This Issue
-
Next Article
Hausdorff dimension for ergodic measures of interval exchange transformations
Almost-everywhere convergence and polynomials
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 |
  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 $\Gamma$.
[1] |
Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363 |
[2] |
Andreas Koutsogiannis. Multiple ergodic averages for tempered functions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1177-1205. doi: 10.3934/dcds.2020314 |
[3] |
Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems & Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044 |
[4] |
Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001 |
[5] |
Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129 |
[6] |
Dmitry Dolgopyat. The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces. Journal of Modern Dynamics, 2020, 16: 351-371. doi: 10.3934/jmd.2020014 |
[7] |
Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129 |
[8] |
Djamel Aaid, Amel Noui, Özen Özer. Piecewise quadratic bounding functions for finding real roots of polynomials. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 63-73. doi: 10.3934/naco.2020015 |
[9] |
He Zhang, John Harlim, Xiantao Li. Estimating linear response statistics using orthogonal polynomials: An RKHS formulation. Foundations of Data Science, 2020, 2 (4) : 443-485. doi: 10.3934/fods.2020021 |
[10] |
Ágota P. Horváth. Discrete diffusion semigroups associated with Jacobi-Dunkl and exceptional Jacobi polynomials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021002 |
[11] |
Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $. Networks & Heterogeneous Media, 2020 doi: 10.3934/nhm.2020033 |
[12] |
Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020395 |
[13] |
Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267 |
[14] |
Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240 |
2019 Impact Factor: 0.465
Tools
Metrics
Other articles
by authors
[Back to Top]