American Institute of Mathematical Sciences

Advanced
November  2007, 18(4): 817-827. doi: 10.3934/dcds.2007.18.817

The return times set and mixing for measure preserving transformations

Rui Kuang 1, and  Xiangdong Ye 1,

1. 

Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, China

Received  March 2006 Revised  May 2007 Published  May 2007

In this paper the relationship between the return times set andseveral mixing properties in measure-theoretical dynamical systems(MDS) is investigated. For an MDS $T$ on a Lebesgue space$(X,$ß,$\mu)$, let ß$^+=\{B\in$ ß$:\mu(B)>0\}$ and$N(A,B)=\{n\in Z_+: \mu(A\cap T^{-n}B)>0\}$ for $A, B\in$ß$^+$. It turns out that $T$ is ergodic iff$N(A,B)$≠$\emptyset$ iff $N(A,B)$ is syndetic; $T$ is weaklymixing iff the lower Banach density of $N(A,B)$ is $1$ iff $N(A,B)$is thick; and $T$ is mildly mixing iff $N(A,B)$ is an $ IP^ * $-set iff$N(A,B)$ is an $(IP-IP)^*$-set for all $A,B\in$ ß$^+$ ifffor each $IP$-set $F$ and $A\in$ß$^+$, $\mu(\bigcup_{n\in{F}}T^{-n}A)=1$. Finally, it is shown that $T$ is intermixing iff$N(A,B)$ is cofinite for all $A,B\in$ß$^+$.
Keywords: density, intermixing, ergodicity, mild mixing, weak mixing, strong mixing, set of return times..
Mathematics Subject Classification: 37A25,37A05; Secondary: 54H2.
Citation: Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817
[1]

Yang Cao, Song Shao. Topological mild mixing of all orders along polynomials. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1163-1184. doi: 10.3934/dcds.2021150
[2]

Krzysztof Frączek, Leonid Polterovich. Growth and mixing. Journal of Modern Dynamics, 2008, 2 (2) : 315-338. doi: 10.3934/jmd.2008.2.315
[3]

Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175
[4]

Oliver Knill. Singular continuous spectrum and quantitative rates of weak mixing. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 33-42. doi: 10.3934/dcds.1998.4.33
[5]

Ethan M. Ackelsberg. Rigidity, weak mixing, and recurrence in abelian groups. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1669-1705. doi: 10.3934/dcds.2021168
[6]

Krzysztof Frączek, M. Lemańczyk, E. Lesigne. Mild mixing property for special flows under piecewise constant functions. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 691-710. doi: 10.3934/dcds.2007.19.691
[7]

Asaf Katz. On mixing and sparse ergodic theorems. Journal of Modern Dynamics, 2021, 17: 1-32. doi: 10.3934/jmd.2021001
[8]

Anthony Quas, Terry Soo. Weak mixing suspension flows over shifts of finite type are universal. Journal of Modern Dynamics, 2012, 6 (4) : 427-449. doi: 10.3934/jmd.2012.6.427
[9]

Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35
[10]

Krzysztof Frączek, Mariusz Lemańczyk. Ratner's property and mild mixing for special flows over two-dimensional rotations. Journal of Modern Dynamics, 2010, 4 (4) : 609-635. doi: 10.3934/jmd.2010.4.609
[11]

Mariusz Lemańczyk. Ergodicity, mixing, Ratner's properties and disjointness for classical flows: On the research of Corinna Ulcigrai. Journal of Modern Dynamics, 2022, 18: 103-130. doi: 10.3934/jmd.2022005
[12]

Lidong Wang, Xiang Wang, Fengchun Lei, Heng Liu. Mixing invariant extremal distributional chaos. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6533-6538. doi: 10.3934/dcds.2016082
[13]

A. Crannell. A chaotic, non-mixing subshift. Conference Publications, 1998, 1998 (Special) : 195-202. doi: 10.3934/proc.1998.1998.195
[14]

Zhi Lin, Katarína Boďová, Charles R. Doering. Models & measures of mixing & effective diffusion. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 259-274. doi: 10.3934/dcds.2010.28.259
[15]

Nir Avni. Spectral and mixing properties of actions of amenable groups. Electronic Research Announcements, 2005, 11: 57-63.

[16]

Richard Miles, Thomas Ward. A directional uniformity of periodic point distribution and mixing. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1181-1189. doi: 10.3934/dcds.2011.30.1181
[17]

Hadda Hmili. Non topologically weakly mixing interval exchanges. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1079-1091. doi: 10.3934/dcds.2010.27.1079
[18]

Ralf Spatzier, Lei Yang. Exponential mixing and smooth classification of commuting expanding maps. Journal of Modern Dynamics, 2017, 11: 263-312. doi: 10.3934/jmd.2017012
[19]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547
[20]

Matúš Dirbák. Minimal skew products with hypertransitive or mixing properties. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1657-1674. doi: 10.3934/dcds.2012.32.1657

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (157)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy