April  2021, 41(4): 1649-1665. 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  April 2021 Early access  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 and Continuous Dynamical Systems, 2021, 41 (4) : 1649-1665. 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.

[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.

[3]

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

[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.

[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.

[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.

[7]

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

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

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

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

[17]

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

[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.

[19]

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

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.

[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.

[3]

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

[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.

[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.

[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.

[7]

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

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

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

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

[17]

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

[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.

[19]

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

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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