American Institute of Mathematical Sciences

Advanced
October  2005, 13(5): 1305-1324. doi: 10.3934/dcds.2005.13.1305

Polymorphisms, Markov processes, quasi-similarity

A. M. Vershik 1,

1. 

St. Petersburg Department of Steklov Institute of Mathematics, 27 Fontanka, St. Petersburg, 191023, Russian Federation

Received  October 2004 Revised  March 2005 Published  September 2005

In this paper we develop the theory of polymorphisms of measure spaces, which is a generalization of the theory of measure-preserving transformations. We describe the main notions and discuss relations to the theory of Markov processes, operator theory, ergodic theory, etc. We formulate the important notion of quasi-similarity and consider quasi-similarity between polymorphisms and automorphisms.

The question is as follows: is it possible to have a quasi-similarity between a measure-preserving automorphism $T$ and a polymorphism $\Pi$ (that is not an automorphism)? In less definite terms: what kind of equivalence can exist between deterministic and random (Markov) dynamical systems? We give the answer: every nonmixing prime polymorphism is quasi-similar to an automorphism with positive entropy, and every $K$-automorphism $T$ is quasi-similar to a polymorphism $\Pi$ that is a special random perturbation of the automorphism $T$.
Keywords: Polymorphisms, Markov maps, perturbations., quasi-similarity.
Mathematics Subject Classification: Primary: 37A05, 47A45, 60J27; Secondary: 37A25,37H10, 47A4.
Citation: A. M. Vershik. Polymorphisms, Markov processes, quasi-similarity. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1305-1324. doi: 10.3934/dcds.2005.13.1305
[1]

Haritha C, Nikita Agarwal. Product of expansive Markov maps with hole. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5743-5774. doi: 10.3934/dcds.2019252
[2]

B. Fernandez, E. Ugalde, J. Urías. Spectrum of dimensions for Poincaré recurrences of Markov maps. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 835-849. doi: 10.3934/dcds.2002.8.835
[3]

Xu Zhang, Yuming Shi, Guanrong Chen. Coupled-expanding maps under small perturbations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1291-1307. doi: 10.3934/dcds.2011.29.1291
[4]

Yujun Zhu. Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 869-882. doi: 10.3934/dcds.2014.34.869
[5]

Marco Lenci. Uniformly expanding Markov maps of the real line: Exactness and infinite mixing. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3867-3903. doi: 10.3934/dcds.2017163
[6]

Manuela Giampieri, Stefano Isola. A one-parameter family of analytic Markov maps with an intermittency transition. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 115-136. doi: 10.3934/dcds.2005.12.115
[7]

Qihuai Liu, Dingbian Qian, Zhiguo Wang. Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1537-1550. doi: 10.3934/dcdsb.2012.17.1537
[8]

Paweł Góra, Abraham Boyarsky. Stochastic perturbations and Ulam's method for W-shaped maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1937-1944. doi: 10.3934/dcds.2013.33.1937
[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]

Jie Liu, Jianguo Si. Invariant tori of a nonlinear Schrödinger equation with quasi-periodically unbounded perturbations. Communications on Pure & Applied Analysis, 2017, 16 (1) : 25-68. doi: 10.3934/cpaa.2017002
[11]

Jose F. Alves; Stefano Luzzatto and Vilton Pinheiro. Markov structures for non-uniformly expanding maps on compact manifolds in arbitrary dimension. Electronic Research Announcements, 2003, 9: 26-31.

[12]

Àlex Haro, Rafael de la Llave. A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1261-1300. doi: 10.3934/dcdsb.2006.6.1261
[13]

Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Superstable periodic orbits of 1d maps under quasi-periodic forcing and reducibility loss. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 589-597. doi: 10.3934/dcds.2014.34.589
[14]

Àngel Jorba, Joan Carles Tatjer. A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 537-567. doi: 10.3934/dcdsb.2008.10.537
[15]

Vassilis G. Papanicolaou, Kyriaki Vasilakopoulou. Similarity solutions of a multidimensional replicator dynamics integrodifferential equation. Journal of Dynamics & Games, 2016, 3 (1) : 51-74. doi: 10.3934/jdg.2016003
[16]

Jose Carlos Camacho, Maria de los Santos Bruzon. Similarity reductions of a nonlinear model for vibrations of beams. Conference Publications, 2015, 2015 (special) : 176-184. doi: 10.3934/proc.2015.0176
[17]

Boran Hu, Zehui Cheng, Zhangbing Zhou. Web services recommendation leveraging semantic similarity computing. Mathematical Foundations of Computing, 2018, 1 (2) : 101-119. doi: 10.3934/mfc.2018006
[18]

Palle Jorgensen, Feng Tian. Dynamical properties of endomorphisms, multiresolutions, similarity and orthogonality relations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2307-2348. doi: 10.3934/dcdss.2019146
[19]

Rogelio Valdez. Self-similarity of the Mandelbrot set for real essentially bounded combinatorics. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 897-922. doi: 10.3934/dcds.2006.16.897
[20]

Ricardo Miranda Martins, Marco Antonio Teixeira. On the similarity of Hamiltonian and reversible vector fields in 4D. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1257-1266. doi: 10.3934/cpaa.2011.10.1257

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences