doi: 10.3934/dcds.2020335

Recurrence for measurable semigroup actions

Institute for Information Transmission Problems RAS (Kharkevich Institute), National Research University "Higher School of Economics", Moscow, Russia

Received  January 2020 Revised  August 2020 Published  October 2020

We study qualitative properties of the set of recurrent points of finitely generated free semigroups of measurable maps. In the case of a single generator the classical Poincare recurrence theorem shows that these properties are closely related to the presence of an invariant measure. Curious, but otherwise it turns out to be possible that almost all points are recurrent, while there is an wandering set of positive (non-invariant) measure. For a general semigroup the assumption about the common invariant measure for all generators looks somewhat unnatural (despite being widely used). Instead we give abstract conditions (of conservativity type) for this problem and propose a weaker version of the recurrent property. Technically, the problem is reduced to the analysis of the recurrence of a specially constructed Markov process. Questions of inheritance of the recurrence property from the semigroup generators to the entire semigroup and vice versa are studied in detail and we demonstrate that this inheritance might be rather unexpected.

Citation: Michael Blank. Recurrence for measurable semigroup actions. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020335
References:
[1]

M. Blank, Topological and metric recurrence for general Markov chains, Moscow Math. J., 19 (2019), 37-50.  doi: 10.17323/1609-4514-2019-19-1-37-50.  Google Scholar

[2]

M. Blank, Perron-Frobenius spectrum for random maps and its approximation, Moscow Math. J., 1 (2001), 315-344.  doi: 10.17323/1609-4514-2001-1-3-315-344.  Google Scholar

[3]

M. Boshernitzan, Quantitative recurrence results, Invent. Math., 113 (1993), 617-631.  doi: 10.1007/BF01244320.  Google Scholar

[4]

M. BoshernitzanN. Frantzikinakis and M. Wierdl, Under recurrence in the Khintchine recurrence theorem, Isr. J. Math., 222 (2017), 815-840.  doi: 10.1007/s11856-017-1606-8.  Google Scholar

[5]

M. CarvalhoF. B. Rodrigues and P. Varandas, Quantitative recurrence for free semigroup actions, Nonlinearity, 31 (2018), 864-886.  doi: 10.1088/1361-6544/aa999f.  Google Scholar

[6]

L. P. Cornfeld, S. V. Fomin and Y. G. Sinai, Ergodic Theory, New York: Springer-Verlag, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[7]

W. Feller, An Introduction to Probability Theory and its Applications, V.1, Wiley, 1966.  Google Scholar

[8] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, 1981.   Google Scholar
[9]

H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions, Anal. Math., 31 (1977), 204-256.  doi: 10.1007/BF02813304.  Google Scholar

[10]

P. R. Halmos, Invariant Measures, Annals of Mathematics, 48 (1947), 735-754.  doi: 10.2307/1969138.  Google Scholar

[11]

Kim Dong Han, Quantitative recurrence properties for group actions, Nonlinearity, 22 (2009), 1-9.  doi: 10.1088/0951-7715/22/1/001.  Google Scholar

[12]

U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics 6, de Gruyter, Berlin-New York, 1985. doi: 10.1515/9783110844641.  Google Scholar

[13]

R. McCutcheon, N. Frantzikinakis, Ergodic Theory: Recurrence, Encyclopedia of Complexity and Systems Science 2007. doi: 10.1007/978-0-387-30440-3_184.  Google Scholar

[14]

S.P. Meyn, R.L. Tweedie, Markov Chains and Stochastic Stability, Springer, New York, 1993. doi: 10.1007/978-1-4471-3267-7.  Google Scholar

[15] Z. Nitecki, Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms, Cambridge (Mass.): The MIT Press, 1971.   Google Scholar
[16]

S. Orey, Recurrent Markov chains, Pacific Journal of Mathematics, 9 (1959), 806-827.  doi: 10.2140/pjm.1959.9.805.  Google Scholar

[17]

V. Spitzer, Principles of Random Walk, Springer-Verlag New York, 1964. doi: 10.1007/978-1-4757-4229-9.  Google Scholar

[18]

A. M. Vershik, What does a typical Markov operator look like?, St. Petersburg Math. J., 17 (2006), 763-772.  doi: 10.1090/S1061-0022-06-00928-9.  Google Scholar

[19]

L.-S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188.  doi: 10.1007/BF02808180.  Google Scholar

show all references

References:
[1]

M. Blank, Topological and metric recurrence for general Markov chains, Moscow Math. J., 19 (2019), 37-50.  doi: 10.17323/1609-4514-2019-19-1-37-50.  Google Scholar

[2]

M. Blank, Perron-Frobenius spectrum for random maps and its approximation, Moscow Math. J., 1 (2001), 315-344.  doi: 10.17323/1609-4514-2001-1-3-315-344.  Google Scholar

[3]

M. Boshernitzan, Quantitative recurrence results, Invent. Math., 113 (1993), 617-631.  doi: 10.1007/BF01244320.  Google Scholar

[4]

M. BoshernitzanN. Frantzikinakis and M. Wierdl, Under recurrence in the Khintchine recurrence theorem, Isr. J. Math., 222 (2017), 815-840.  doi: 10.1007/s11856-017-1606-8.  Google Scholar

[5]

M. CarvalhoF. B. Rodrigues and P. Varandas, Quantitative recurrence for free semigroup actions, Nonlinearity, 31 (2018), 864-886.  doi: 10.1088/1361-6544/aa999f.  Google Scholar

[6]

L. P. Cornfeld, S. V. Fomin and Y. G. Sinai, Ergodic Theory, New York: Springer-Verlag, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[7]

W. Feller, An Introduction to Probability Theory and its Applications, V.1, Wiley, 1966.  Google Scholar

[8] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, 1981.   Google Scholar
[9]

H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions, Anal. Math., 31 (1977), 204-256.  doi: 10.1007/BF02813304.  Google Scholar

[10]

P. R. Halmos, Invariant Measures, Annals of Mathematics, 48 (1947), 735-754.  doi: 10.2307/1969138.  Google Scholar

[11]

Kim Dong Han, Quantitative recurrence properties for group actions, Nonlinearity, 22 (2009), 1-9.  doi: 10.1088/0951-7715/22/1/001.  Google Scholar

[12]

U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics 6, de Gruyter, Berlin-New York, 1985. doi: 10.1515/9783110844641.  Google Scholar

[13]

R. McCutcheon, N. Frantzikinakis, Ergodic Theory: Recurrence, Encyclopedia of Complexity and Systems Science 2007. doi: 10.1007/978-0-387-30440-3_184.  Google Scholar

[14]

S.P. Meyn, R.L. Tweedie, Markov Chains and Stochastic Stability, Springer, New York, 1993. doi: 10.1007/978-1-4471-3267-7.  Google Scholar

[15] Z. Nitecki, Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms, Cambridge (Mass.): The MIT Press, 1971.   Google Scholar
[16]

S. Orey, Recurrent Markov chains, Pacific Journal of Mathematics, 9 (1959), 806-827.  doi: 10.2140/pjm.1959.9.805.  Google Scholar

[17]

V. Spitzer, Principles of Random Walk, Springer-Verlag New York, 1964. doi: 10.1007/978-1-4757-4229-9.  Google Scholar

[18]

A. M. Vershik, What does a typical Markov operator look like?, St. Petersburg Math. J., 17 (2006), 763-772.  doi: 10.1090/S1061-0022-06-00928-9.  Google Scholar

[19]

L.-S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188.  doi: 10.1007/BF02808180.  Google Scholar

Figure 1.  Graphs of the maps $ T_i $ in the Example 5
Figure 2.  Graphs of the maps $ T_i $ in the Example 6
Figure 3.  Graphs of the maps $ T_i $ in the Example 7
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