On uniformly recurrent motions of topological semigroup actions

Previous Article
DCDS Home
This Issue
Next Article
Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data

2016, 36(6): 2931-2944. doi: 10.3934/dcds.2016.36.2931

On uniformly recurrent motions of topological semigroup actions

1.	Department of Mathematics, Nanjing University, Nanjing 210093, China
2.	Department of Mathematics, Nanjing University, Nanjing, 210093

Received October 2014 Revised October 2015 Published December 2015

Abstract
Related Papers

Let G ↷ X be a topological action of a topological semigroup $G$ on a compact metric space $X$. We show in this paper that for any given point $x$ in $X$, the following two properties that both approximate to periodicity are equivalent to each other:
$\bullet$ For any neighborhood $U$ of $x$, the return times set $\{g\in G ： gx\in U\}$ is syndetic of Furstenburg in $G$.
$\bullet$ Given any $\varepsilon>0$, there exists a finite subset $K$ of $G$ such that for each $g$ in $G$, the $\varepsilon$-neighborhood of the orbit-arc $K[gx]$ contains the entire orbit $G[x]$.
This is a generalization of a classical theorem of Birkhoff for the case where $G=\mathbb{R}$ or $\mathbb{Z}$. In addition, a counterexample is constructed to this statement, while $X$ is merely a complete but not locally compact metric space.

Keywords: uniformly recurrent point, almost periodic point., $G$-space.

Mathematics Subject Classification: 37B05, 37B2.

Citation: Bin Chen, Xiongping Dai. On uniformly recurrent motions of topological semigroup actions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2931-2944. doi: 10.3934/dcds.2016.36.2931

References:

[1]	J. Egawa, A characterization of regularly almost periodic minimal flows,, Proc. Japan Acad. Ser. A, 71 (1995), 225. doi: 10.3792/pjaa.71.225.
[2]	H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,, Princeton University Press, (1981).
[3]	W. H. Gottschalk, Almost periodic points with respect to transformation semi-groups,, Annals of Math., 47 (1946), 762. doi: 10.2307/1969233.
[4]	W. H. Gottschalk, A survey of minimal sets,, Ann. Inst. Fourier, 14 (1964), 53. doi: 10.5802/aif.160.
[5]	W. H. Gottschalk and G. A. Hedlund, Topological Dynamics,, Amer. Math. Soc. Coll. Publ., (1955).
[6]	A. Miller and J. Rosenblatt, Characterizations of regular almost periodicity in compact minimal abelian flows,, Trans. Amer. Math. Soc., 356* (2004), 4909. doi: 10.1090/S0002-9947-04-03538-X.*
[7]	D. Montgomery, Almost periodic transformation groups,, Trans. Amer. Math. Soc., 42 (1937), 322. doi: 10.1090/S0002-9947-1937-1501924-0.
[8]	V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations,, Princeton University Press, (1960).
[9]	A. Weil, L'Integration Dans Les Groupes Topologiques et Ses Applications,, Actualitiés scientifiques, (1938).

show all references

References:

[1]	J. Egawa, A characterization of regularly almost periodic minimal flows,, Proc. Japan Acad. Ser. A, 71 (1995), 225. doi: 10.3792/pjaa.71.225.
[2]	H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,, Princeton University Press, (1981).
[3]	W. H. Gottschalk, Almost periodic points with respect to transformation semi-groups,, Annals of Math., 47 (1946), 762. doi: 10.2307/1969233.
[4]	W. H. Gottschalk, A survey of minimal sets,, Ann. Inst. Fourier, 14 (1964), 53. doi: 10.5802/aif.160.
[5]	W. H. Gottschalk and G. A. Hedlund, Topological Dynamics,, Amer. Math. Soc. Coll. Publ., (1955).
[6]	A. Miller and J. Rosenblatt, Characterizations of regular almost periodicity in compact minimal abelian flows,, Trans. Amer. Math. Soc., 356* (2004), 4909. doi: 10.1090/S0002-9947-04-03538-X.*
[7]	D. Montgomery, Almost periodic transformation groups,, Trans. Amer. Math. Soc., 42 (1937), 322. doi: 10.1090/S0002-9947-1937-1501924-0.
[8]	V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations,, Princeton University Press, (1960).
[9]	A. Weil, L'Integration Dans Les Groupes Topologiques et Ses Applications,, Actualitiés scientifiques, (1938).

[1]	Álvaro Castañeda, Gonzalo Robledo. Almost reducibility of linear difference systems from a spectral point of view. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1977-1988. doi: 10.3934/cpaa.2017097
[2]	Richard Miles, Thomas Ward. A directional uniformity of periodic point distribution and mixing. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1181-1189. doi: 10.3934/dcds.2011.30.1181
[3]	Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807
[4]	Hannelore Lisei, Radu Precup, Csaba Varga. A Schechter type critical point result in annular conical domains of a Banach space and applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3775-3789. doi: 10.3934/dcds.2016.36.3775
[5]	Jiao Chen, Weike Wang. The point-wise estimates for the solution of damped wave equation with nonlinear convection in multi-dimensional space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 307-330. doi: 10.3934/cpaa.2014.13.307
[6]	Paolo Perfetti. Fixed point theorems in the Arnol'd model about instability of the action-variables in phase-space. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 379-391. doi: 10.3934/dcds.1998.4.379
[7]	G. A. Swarup. On the cut point conjecture. Electronic Research Announcements, 1996, 2: 98-100.
[8]	Marek Rychlik. The Equichordal Point Problem. Electronic Research Announcements, 1996, 2: 108-123.
[9]	Junxiang Xu. On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2593-2619. doi: 10.3934/dcds.2013.33.2593
[10]	Wenhua Qiu, Jianguo Si. On small perturbation of four-dimensional quasi-periodic system with degenerate equilibrium point. Communications on Pure & Applied Analysis, 2015, 14 (2) : 421-437. doi: 10.3934/cpaa.2015.14.421
[11]	Ralf Kirsch, Sergej Rjasanow. The uniformly heated inelastic Boltzmann equation in Fourier space. Kinetic & Related Models, 2010, 3 (3) : 445-456. doi: 10.3934/krm.2010.3.445
[12]	Zvia Agur, L. Arakelyan, P. Daugulis, Y. Ginosar. Hopf point analysis for angiogenesis models. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 29-38. doi: 10.3934/dcdsb.2004.4.29
[13]	Philip N. J. Eagle, Steven D. Galbraith, John B. Ong. Point compression for Koblitz elliptic curves. Advances in Mathematics of Communications, 2011, 5 (1) : 1-10. doi: 10.3934/amc.2011.5.1
[14]	Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297
[15]	Giuseppe Marino, Hong-Kun Xu. Convergence of generalized proximal point algorithms. Communications on Pure & Applied Analysis, 2004, 3 (4) : 791-808. doi: 10.3934/cpaa.2004.3.791
[16]	Leong-Kwan Li, Sally Shao. Convergence analysis of the weighted state space search algorithm for recurrent neural networks. Numerical Algebra, Control & Optimization, 2014, 4 (3) : 193-207. doi: 10.3934/naco.2014.4.193
[17]	Juan Sánchez, Marta Net, José M. Vega. Amplitude equations close to a triple-(+1) bifurcation point of D4-symmetric periodic orbits in O(2)-equivariant systems. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1357-1380. doi: 10.3934/dcdsb.2006.6.1357
[18]	Jianqin Zhou, Wanquan Liu, Xifeng Wang. Complete characterization of the first descent point distribution for the k-error linear complexity of 2ⁿ-periodic binary sequences. Advances in Mathematics of Communications, 2017, 11 (3) : 429-444. doi: 10.3934/amc.2017036
[19]	Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703
[20]	Peter Giesl, Martin Rasmussen. A note on almost periodic variational equations. Communications on Pure & Applied Analysis, 2011, 10 (3) : 983-994. doi: 10.3934/cpaa.2011.10.983

2016 Impact Factor: 1.099

American Institute of Mathematical Sciences