American Institute of Mathematical Sciences

Advanced
June  2009, 2(2): 221-238. doi: 10.3934/dcdss.2009.2.221

Measured topological orbit and Kakutani equivalence

Andres del Junco 1, , Daniel J. Rudolph 2, and  Benjamin Weiss 3,

1. 

Department of Mathematics, University of Toronto, Toronto, Ontario, Canada
2. 

Department of Mathematics, Colorado State University, Fort Collins, CO 80523, United States
3. 

Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904

Received  February 2008 Revised  October 2008 Published  April 2009

Suppose $X$ and $Y$ are Polish spaces each endowed with Borel probability measures $\mu$ and $\nu$. We call these Polish probability spaces. We say a map $\phi$ is a nearly continuous if there are measurable subsets $X_0\subseteq X$ and $Y_0\subseteq Y$, each of full measure, and $\phi:X_0\to Y_0$ is measure-preserving and continuous in the relative topologies on these subsets. We show that this is a natural context to study morphisms between ergodic homeomorphisms of Polish probability spaces. In previous work such maps have been called almost continuous or finitary. We propose the name measured topological dynamics for this area of study. Suppose one has measure-preserving and ergodic maps $T$ and $S$ acting on $X$ and $Y$ respectively. Suppose $\phi$ is a measure-preserving bijection defined between subsets of full measure on these two spaces. Our main result is that such a $\phi$ can always be regularized in the following sense. Both $T$ and $S$ have full groups ($FG(T)$ and $FG(S)$) consisting of those measurable bijections that carry a point to a point on the same orbit. We will show that there exists $f\in FG(T)$ and $h\in FG(S)$ so that $h\phi f$ is nearly continuous. This comes close to giving an alternate proof of the result of del Junco and Şahin, that any two measure-preserving ergodic homeomorphisms of nonatomic Polish probability spaces are continuously orbit equivalent on invariant $G_\delta$ subsets of full measure. One says $T$ and $S$ are evenly Kakutani equivalent if one has an orbit equivalence $\phi$ which restricted to some subset is a conjugacy of the induced maps. Our main result implies that any such measurable Kakutani equivalence can be regularized to a Kakutani equivalence that is nearly continuous. We describe a natural nearly continuous analogue of Kakutani equivalence and prove it strictly stronger than Kakutani equivalence. To do this we introduce a concept of nearly unique ergodicity.
Keywords: Kakutani equivalence, finitary, orbit equivalence, Measured Topological Dynamics.
Mathematics Subject Classification: Primary: 37A05, 37A20 ; Secondary: 37A15, 54H2.
Citation: Andres del Junco, Daniel J. Rudolph, Benjamin Weiss. Measured topological orbit and Kakutani equivalence. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 221-238. doi: 10.3934/dcdss.2009.2.221
[1]

Mrinal Kanti Roychowdhury, Daniel J. Rudolph. Nearly continuous Kakutani equivalence of adding machines. Journal of Modern Dynamics, 2009, 3 (1) : 103-119. doi: 10.3934/jmd.2009.3.103
[2]

Kengo Matsumoto. Continuous orbit equivalence of topological Markov shifts and KMS states on Cuntz–Krieger algebras. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5897-5909. doi: 10.3934/dcds.2020251
[3]

Luis Barreira, Liviu Horia Popescu, Claudia Valls. Generalized exponential behavior and topological equivalence. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3023-3042. doi: 10.3934/dcdsb.2017161
[4]

Giuseppe Buttazzo, Luigi De Pascale, Ilaria Fragalà. Topological equivalence of some variational problems involving distances. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 247-258. doi: 10.3934/dcds.2001.7.247
[5]

Álvaro Castañeda, Pablo González, Gonzalo Robledo. Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line. Communications on Pure & Applied Analysis, 2021, 20 (2) : 511-532. doi: 10.3934/cpaa.2020278
[6]

Keonhee Lee, Kazuhiro Sakai. Various shadowing properties and their equivalence. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 533-540. doi: 10.3934/dcds.2005.13.533
[7]

Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Advances in Mathematics of Communications, 2010, 4 (1) : 69-81. doi: 10.3934/amc.2010.4.69
[8]

Michael C. Sullivan. Invariants of twist-wise flow equivalence. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 475-484. doi: 10.3934/dcds.1998.4.475
[9]

Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Addendum. Advances in Mathematics of Communications, 2011, 5 (3) : 543-546. doi: 10.3934/amc.2011.5.543
[10]

Nguyen Lam. Equivalence of sharp Trudinger-Moser-Adams Inequalities. Communications on Pure & Applied Analysis, 2017, 16 (3) : 973-998. doi: 10.3934/cpaa.2017047
[11]

Zemer Kosloff, Terry Soo. The orbital equivalence of Bernoulli actions and their Sinai factors. Journal of Modern Dynamics, 2021, 17: 145-182. doi: 10.3934/jmd.2021005
[12]

Mike Crampin, David Saunders. Homogeneity and projective equivalence of differential equation fields. Journal of Geometric Mechanics, 2012, 4 (1) : 27-47. doi: 10.3934/jgm.2012.4.27
[13]

Michael C. Sullivan. Invariants of twist-wise flow equivalence. Electronic Research Announcements, 1997, 3: 126-130.

[14]

Kurt Ehlers. Geometric equivalence on nonholonomic three-manifolds. Conference Publications, 2003, 2003 (Special) : 246-255. doi: 10.3934/proc.2003.2003.246
[15]

B. Kaymakcalan, R. Mert, A. Zafer. Asymptotic equivalence of dynamic systems on time scales. Conference Publications, 2007, 2007 (Special) : 558-567. doi: 10.3934/proc.2007.2007.558
[16]

Brett M. Werner. An example of Kakutani equivalent and strong orbit equivalent substitution systems that are not conjugate. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 239-249. doi: 10.3934/dcdss.2009.2.239
[17]

Ricardo Miranda Martins. Formal equivalence between normal forms of reversible and hamiltonian dynamical systems. Communications on Pure & Applied Analysis, 2014, 13 (2) : 703-713. doi: 10.3934/cpaa.2014.13.703
[18]

J. Gwinner. On differential variational inequalities and projected dynamical systems - equivalence and a stability result. Conference Publications, 2007, 2007 (Special) : 467-476. doi: 10.3934/proc.2007.2007.467
[19]

Kan Jiang, Lifeng Xi, Shengnan Xu, Jinjin Yang. Isomorphism and bi-Lipschitz equivalence between the univoque sets. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6089-6114. doi: 10.3934/dcds.2020271
[20]

Stephen McDowall, Plamen Stefanov, Alexandru Tamasan. Gauge equivalence in stationary radiative transport through media with varying index of refraction. Inverse Problems & Imaging, 2010, 4 (1) : 151-167. doi: 10.3934/ipi.2010.4.151

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences