doi: 10.3934/dcds.2021168
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Rigidity, weak mixing, and recurrence in abelian groups

Department of Mathematics, Ohio State University, Columbus, OH 43210, USA

Received  July 2020 Revised  September 2021 Early access November 2021

The focus of this paper is the phenomenon of rigidity for measure-preserving actions of countable discrete abelian groups and its interactions with weak mixing and recurrence. We prove that results about $ \mathbb{Z} $-actions extend to this setting:

1. If $ (a_n) $ is a rigidity sequence for an ergodic measure-preserving system, then it is a rigidity sequence for some weakly mixing system.

2. There exists a sequence $ (r_n) $ such that every translate is both a rigidity sequence and a set of recurrence.

The first of these results was shown for $ \mathbb{Z} $-actions by Adams [1], Fayad and Thouvenot [20], and Badea and Grivaux [2]. The latter was established in $ \mathbb{Z} $ by Griesmer [23]. While techniques for handling $ \mathbb{Z} $-actions play a key role in our proofs, additional ideas must be introduced for dealing with groups with multiple generators.

As an application of our results, we give several new constructions of rigidity sequences in torsion groups. Some of these are parallel to examples of rigidity sequences in $ \mathbb{Z} $, while others exhibit new phenomena.

Citation: Ethan M. Ackelsberg. Rigidity, weak mixing, and recurrence in abelian groups. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021168
References:
[1]

T. Adams, Tower multiplexing and slow weak mixing, Colloq. Math., 138 (2015), 47-72.  doi: 10.4064/cm138-1-4.  Google Scholar

[2]

C. Badea and S. Grivaux, Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences, Comment. Math. Helv., 95 (2020), 99-127.  doi: 10.4171/CMH/482.  Google Scholar

[3]

C. BadeaS. Grivaux and Étienne Matheron, Rigidity sequences, Kazhdan sets and group topologies on the integers, J. Anal. Math., 143 (2021), 313-347.  doi: 10.1007/s11854-021-0165-4.  Google Scholar

[4]

G. BarbieriD. DikranjanC. Milan and H. Weber, Answer to Raczkowski's question on convergent sequences of integers, Topology Appl., 132 (2003), 89-101.  doi: 10.1016/S0166-8641(02)00366-8.  Google Scholar

[5]

P. T. Bateman and A. L. Duquette, The analogue of the Pisot-Vijayaraghavan numbers in fields of formal power series, Illinois J. Math., 6 (1962), 594-606.   Google Scholar

[6]

V. Bergelson, Ergodic theory and diophantine problems, In Topics in Symbolic Dynamics and Applications (Temuco, 1997), London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 279 (2000), 167–205. Google Scholar

[7]

V. Bergelson, Combinatorial and diophantine applications of ergodic theory, Handbook of Dynamical Systems., 1 (2006), 745-869.   Google Scholar

[8]

V. BergelsonA. del JuncoM. Lemańczyk and J. Rosenblatt, Rigidity and non-recurrence along sequences, Ergodic Theory Dynam. Systems, 34 (2014), 1464-1502.  doi: 10.1017/etds.2013.5.  Google Scholar

[9]

V. Bergelson and D. Glasscock, Multiplicative richness of additively large sets in $\mathbb{Z}^d$, J. Algebra, 503 (2018), 67-103.  doi: 10.1016/j.jalgebra.2018.01.032.  Google Scholar

[10]

V. Bergelson and I. J. Håland, Sets of recurrence and generalized polynomials, In Convergence in Ergodic Theory and Probability (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., de Gruyter, Berlin, 5 (1996), 91–110.  Google Scholar

[11]

V. Bergelson and A. Leibman, A Weyl-type equidistribution theorem in finite characteristic, Adv. Math., 289 (2016), 928-950.  doi: 10.1016/j.aim.2015.11.027.  Google Scholar

[12]

V. Bergelson and J. Rosenblatt, Mixing actions of groups, Illinois J. Math., 32 (1988), 65-80.   Google Scholar

[13]

P. Billingsley, Convergence of Probability Measures, 2$^{nd}$ edition, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[14]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinaĭ, Ergodic Theory, vol. 245 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

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R. Diestel, Graph Theory, 5$^{th}$ edition, vol. 173 of Graduate Texts in Mathematics, Springer, Berlin, 2017. doi: 10.1007/978-3-662-53622-3.  Google Scholar

[16]

D. DikranjanC. Milan and A. Tonolo, A characterization of the maximally almost periodic abelian groups, J. Pure Appl. Algebra, 197 (2005), 23-41.  doi: 10.1016/j.jpaa.2004.08.021.  Google Scholar

[17]

T. DownarowiczD. Huczek and G. Zhang, Tilings of amenable groups, J. Reine Angew. Math., 747 (2019), 277-298.  doi: 10.1515/crelle-2016-0025.  Google Scholar

[18]

H. A. Dye, On the ergodic mixing theorem, Trans. Amer. Math. Soc., 118 (1965), 123-130.  doi: 10.1090/S0002-9947-1965-0174705-8.  Google Scholar

[19]

B. Fayad and A. Kanigowski, Rigidity times for a weakly mixing dynamical system which are not rigidity times for any irrational rotation, Ergodic Theory Dynam. Systems, 35 (2015), 2529-2534.  doi: 10.1017/etds.2014.40.  Google Scholar

[20]

B. Fayad and J.-P. Thouvenot, On the convergence to 0 of $m_n\xi $ mod 1, Acta Arith., 165 (2014), 327-332.  doi: 10.4064/aa165-4-2.  Google Scholar

[21]

A. H. Forrest, Recurrence in Dynamical Systems: A Combinatorial Approach, Ph.D thesis, The Ohio State University, 1990.  Google Scholar

[22] L. Fuchs, Infinite Abelian Groups. Vol. I, vol. 36 of Pure and Applies Mathematics, Academic Press, New York-London, 1970.   Google Scholar
[23]

J. T. Griesmer, Recurrence, rigidity, and popular differences, Ergodic Theory Dynam. Systems, 39 (2019), 1299-1316.  doi: 10.1017/etds.2017.71.  Google Scholar

[24]

P. R. Halmos, In general a measure preserving transformation is mixing, Ann. of Math. (2), 45 (1944), 786-792.  doi: 10.2307/1969304.  Google Scholar

[25]

E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. I, 2$^{nd}$ edition, vol. 115 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin-New York, 1979.  Google Scholar

[26]

E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. II, vol. 152 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin-New York, 1970.  Google Scholar

[27] S. Janson, Gaussian Hilbert Spaces, vol. 129 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1997.  doi: 10.1017/CBO9780511526169.  Google Scholar
[28]

A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, Handbook of Dynamical Systems., 1B (2006), 649-743.  doi: 10.1016/S1874-575X(06)80036-6.  Google Scholar

[29]

A. S. Kechris, Global Aspects of Ergodic Group Actions, vol. 160 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/160.  Google Scholar

[30]

B. O. Koopman and J. von Neumann, Dynamical systems of continuous spectra, Proc. Nat. Acad. Sci. U.S.A., 18 (1932), 255-263.  doi: 10.1073/pnas.18.3.255.  Google Scholar

[31]

C. McDiarmid., On the method of bounded differences, Surveys in Combinatorics, 1989 (Norwich, 1989), Cambridge Univ. Press, Cambridge, 141 (1989), 148-188.   Google Scholar

[32]

M. G. Nadkarni, Basic Ergodic Theory, 3$^{rd}$ edition, vol. 6 of Texts and Readings in Mathematics, Hindustan Book Agency, New Delhi, 2013.  Google Scholar

[33]

D. S. Ornstein and B. Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.), 2 (1980), 161-164.  doi: 10.1090/S0273-0979-1980-14702-3.  Google Scholar

[34]

I. V. Protasov and E. G. Zelenyuk, Topologies on abelian groups, Izv. Akad. Nauk SSSR Ser. Mat., 54 (1990), 1090-1107.  doi: 10.1070/IM1991v037n02ABEH002071.  Google Scholar

[35]

W. Rudin, Fourier Analysis on Groups, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. doi: 10.1002/9781118165621.  Google Scholar

[36]

I. Z. Ruzsa, Arithmetical topology, Number Theory, Vol. I (Budapest, 1987), Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam, 51 (1990), 473-504.   Google Scholar

[37]

K. Schmidt, Asymptotic properties of unitary representations and mixing, Proc. London Math. Soc., 48 (1984), 445-460.  doi: 10.1112/plms/s3-48.3.445.  Google Scholar

[38]

K. Schmidt and P. Walters, Mildly mixing actions of locally compact groups, Proc. London Math. Soc., 45 (1982), 506-518.  doi: 10.1112/plms/s3-45.3.506.  Google Scholar

[39]

W. M. Schmidt, On continued fractions and Diophantine approximation in power series fields, Acta Arith., 95 (2000), 139-166.  doi: 10.4064/aa-95-2-139-166.  Google Scholar

[40]

P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[41]

B. Weiss, Monotileable amenable groups, Amer. Math. Soc. Transl. Ser. 2, 202 (2001), 257-262.  doi: 10.1090/trans2/202/18.  Google Scholar

show all references

References:
[1]

T. Adams, Tower multiplexing and slow weak mixing, Colloq. Math., 138 (2015), 47-72.  doi: 10.4064/cm138-1-4.  Google Scholar

[2]

C. Badea and S. Grivaux, Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences, Comment. Math. Helv., 95 (2020), 99-127.  doi: 10.4171/CMH/482.  Google Scholar

[3]

C. BadeaS. Grivaux and Étienne Matheron, Rigidity sequences, Kazhdan sets and group topologies on the integers, J. Anal. Math., 143 (2021), 313-347.  doi: 10.1007/s11854-021-0165-4.  Google Scholar

[4]

G. BarbieriD. DikranjanC. Milan and H. Weber, Answer to Raczkowski's question on convergent sequences of integers, Topology Appl., 132 (2003), 89-101.  doi: 10.1016/S0166-8641(02)00366-8.  Google Scholar

[5]

P. T. Bateman and A. L. Duquette, The analogue of the Pisot-Vijayaraghavan numbers in fields of formal power series, Illinois J. Math., 6 (1962), 594-606.   Google Scholar

[6]

V. Bergelson, Ergodic theory and diophantine problems, In Topics in Symbolic Dynamics and Applications (Temuco, 1997), London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 279 (2000), 167–205. Google Scholar

[7]

V. Bergelson, Combinatorial and diophantine applications of ergodic theory, Handbook of Dynamical Systems., 1 (2006), 745-869.   Google Scholar

[8]

V. BergelsonA. del JuncoM. Lemańczyk and J. Rosenblatt, Rigidity and non-recurrence along sequences, Ergodic Theory Dynam. Systems, 34 (2014), 1464-1502.  doi: 10.1017/etds.2013.5.  Google Scholar

[9]

V. Bergelson and D. Glasscock, Multiplicative richness of additively large sets in $\mathbb{Z}^d$, J. Algebra, 503 (2018), 67-103.  doi: 10.1016/j.jalgebra.2018.01.032.  Google Scholar

[10]

V. Bergelson and I. J. Håland, Sets of recurrence and generalized polynomials, In Convergence in Ergodic Theory and Probability (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., de Gruyter, Berlin, 5 (1996), 91–110.  Google Scholar

[11]

V. Bergelson and A. Leibman, A Weyl-type equidistribution theorem in finite characteristic, Adv. Math., 289 (2016), 928-950.  doi: 10.1016/j.aim.2015.11.027.  Google Scholar

[12]

V. Bergelson and J. Rosenblatt, Mixing actions of groups, Illinois J. Math., 32 (1988), 65-80.   Google Scholar

[13]

P. Billingsley, Convergence of Probability Measures, 2$^{nd}$ edition, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[14]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinaĭ, Ergodic Theory, vol. 245 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[15]

R. Diestel, Graph Theory, 5$^{th}$ edition, vol. 173 of Graduate Texts in Mathematics, Springer, Berlin, 2017. doi: 10.1007/978-3-662-53622-3.  Google Scholar

[16]

D. DikranjanC. Milan and A. Tonolo, A characterization of the maximally almost periodic abelian groups, J. Pure Appl. Algebra, 197 (2005), 23-41.  doi: 10.1016/j.jpaa.2004.08.021.  Google Scholar

[17]

T. DownarowiczD. Huczek and G. Zhang, Tilings of amenable groups, J. Reine Angew. Math., 747 (2019), 277-298.  doi: 10.1515/crelle-2016-0025.  Google Scholar

[18]

H. A. Dye, On the ergodic mixing theorem, Trans. Amer. Math. Soc., 118 (1965), 123-130.  doi: 10.1090/S0002-9947-1965-0174705-8.  Google Scholar

[19]

B. Fayad and A. Kanigowski, Rigidity times for a weakly mixing dynamical system which are not rigidity times for any irrational rotation, Ergodic Theory Dynam. Systems, 35 (2015), 2529-2534.  doi: 10.1017/etds.2014.40.  Google Scholar

[20]

B. Fayad and J.-P. Thouvenot, On the convergence to 0 of $m_n\xi $ mod 1, Acta Arith., 165 (2014), 327-332.  doi: 10.4064/aa165-4-2.  Google Scholar

[21]

A. H. Forrest, Recurrence in Dynamical Systems: A Combinatorial Approach, Ph.D thesis, The Ohio State University, 1990.  Google Scholar

[22] L. Fuchs, Infinite Abelian Groups. Vol. I, vol. 36 of Pure and Applies Mathematics, Academic Press, New York-London, 1970.   Google Scholar
[23]

J. T. Griesmer, Recurrence, rigidity, and popular differences, Ergodic Theory Dynam. Systems, 39 (2019), 1299-1316.  doi: 10.1017/etds.2017.71.  Google Scholar

[24]

P. R. Halmos, In general a measure preserving transformation is mixing, Ann. of Math. (2), 45 (1944), 786-792.  doi: 10.2307/1969304.  Google Scholar

[25]

E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. I, 2$^{nd}$ edition, vol. 115 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin-New York, 1979.  Google Scholar

[26]

E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. II, vol. 152 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin-New York, 1970.  Google Scholar

[27] S. Janson, Gaussian Hilbert Spaces, vol. 129 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1997.  doi: 10.1017/CBO9780511526169.  Google Scholar
[28]

A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, Handbook of Dynamical Systems., 1B (2006), 649-743.  doi: 10.1016/S1874-575X(06)80036-6.  Google Scholar

[29]

A. S. Kechris, Global Aspects of Ergodic Group Actions, vol. 160 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/160.  Google Scholar

[30]

B. O. Koopman and J. von Neumann, Dynamical systems of continuous spectra, Proc. Nat. Acad. Sci. U.S.A., 18 (1932), 255-263.  doi: 10.1073/pnas.18.3.255.  Google Scholar

[31]

C. McDiarmid., On the method of bounded differences, Surveys in Combinatorics, 1989 (Norwich, 1989), Cambridge Univ. Press, Cambridge, 141 (1989), 148-188.   Google Scholar

[32]

M. G. Nadkarni, Basic Ergodic Theory, 3$^{rd}$ edition, vol. 6 of Texts and Readings in Mathematics, Hindustan Book Agency, New Delhi, 2013.  Google Scholar

[33]

D. S. Ornstein and B. Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.), 2 (1980), 161-164.  doi: 10.1090/S0273-0979-1980-14702-3.  Google Scholar

[34]

I. V. Protasov and E. G. Zelenyuk, Topologies on abelian groups, Izv. Akad. Nauk SSSR Ser. Mat., 54 (1990), 1090-1107.  doi: 10.1070/IM1991v037n02ABEH002071.  Google Scholar

[35]

W. Rudin, Fourier Analysis on Groups, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. doi: 10.1002/9781118165621.  Google Scholar

[36]

I. Z. Ruzsa, Arithmetical topology, Number Theory, Vol. I (Budapest, 1987), Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam, 51 (1990), 473-504.   Google Scholar

[37]

K. Schmidt, Asymptotic properties of unitary representations and mixing, Proc. London Math. Soc., 48 (1984), 445-460.  doi: 10.1112/plms/s3-48.3.445.  Google Scholar

[38]

K. Schmidt and P. Walters, Mildly mixing actions of locally compact groups, Proc. London Math. Soc., 45 (1982), 506-518.  doi: 10.1112/plms/s3-45.3.506.  Google Scholar

[39]

W. M. Schmidt, On continued fractions and Diophantine approximation in power series fields, Acta Arith., 95 (2000), 139-166.  doi: 10.4064/aa-95-2-139-166.  Google Scholar

[40]

P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[41]

B. Weiss, Monotileable amenable groups, Amer. Math. Soc. Transl. Ser. 2, 202 (2001), 257-262.  doi: 10.1090/trans2/202/18.  Google Scholar

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