April  2022, 42(4): 1669-1705. doi: 10.3934/dcds.2021168

Rigidity, weak mixing, and recurrence in abelian groups

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

Received  July 2020 Revised  September 2021 Published  April 2022 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 and Continuous Dynamical Systems, 2022, 42 (4) : 1669-1705. 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.

[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.

[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.

[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.

[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. 

[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.

[7]

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

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

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

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

[24]

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

[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.

[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.

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

[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.

[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.

[31]

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

[32]

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

[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.

[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.

[35]

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

[36]

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

[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.

[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.

[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.

[40]

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

[41]

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

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.

[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.

[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.

[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.

[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. 

[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.

[7]

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

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

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

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

[24]

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

[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.

[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.

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

[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.

[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.

[31]

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

[32]

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

[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.

[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.

[35]

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

[36]

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

[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.

[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.

[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.

[40]

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

[41]

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

[1]

Mads R. Bisgaard. Mather theory and symplectic rigidity. Journal of Modern Dynamics, 2019, 15: 165-207. doi: 10.3934/jmd.2019018

[2]

Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757

[3]

Thierry de la Rue. An introduction to joinings in ergodic theory. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 121-142. doi: 10.3934/dcds.2006.15.121

[4]

Andrei Török. Rigidity of partially hyperbolic actions of property (T) groups. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 193-208. doi: 10.3934/dcds.2003.9.193

[5]

David Mieczkowski. The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory. Journal of Modern Dynamics, 2007, 1 (1) : 61-92. doi: 10.3934/jmd.2007.1.61

[6]

Danijela Damjanovic and Anatole Katok. Local rigidity of actions of higher rank abelian groups and KAM method. Electronic Research Announcements, 2004, 10: 142-154.

[7]

Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22

[8]

Cristina Lizana, Vilton Pinheiro, Paulo Varandas. Contribution to the ergodic theory of robustly transitive maps. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 353-365. doi: 10.3934/dcds.2015.35.353

[9]

Michael Hochman. Lectures on dynamics, fractal geometry, and metric number theory. Journal of Modern Dynamics, 2014, 8 (3&4) : 437-497. doi: 10.3934/jmd.2014.8.437

[10]

E. Muñoz Garcia, R. Pérez-Marco. Diophantine conditions in small divisors and transcendental number theory. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1401-1409. doi: 10.3934/dcds.2003.9.1401

[11]

Michael Blank. Recurrence for measurable semigroup actions. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1649-1665. doi: 10.3934/dcds.2020335

[12]

Jingxian Sun, Shouchuan Hu. Flow-invariant sets and critical point theory. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 483-496. doi: 10.3934/dcds.2003.9.483

[13]

Zhenqi Jenny Wang. New cases of differentiable rigidity for partially hyperbolic actions: Symplectic groups and resonance directions. Journal of Modern Dynamics, 2010, 4 (4) : 585-608. doi: 10.3934/jmd.2010.4.585

[14]

Masayuki Asaoka. Local rigidity of homogeneous actions of parabolic subgroups of rank-one Lie groups. Journal of Modern Dynamics, 2015, 9: 191-201. doi: 10.3934/jmd.2015.9.191

[15]

Anatole Katok, Federico Rodriguez Hertz. Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups. Journal of Modern Dynamics, 2010, 4 (3) : 487-515. doi: 10.3934/jmd.2010.4.487

[16]

Mao Okada. Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces. Journal of Modern Dynamics, 2021, 17: 111-143. doi: 10.3934/jmd.2021004

[17]

Danijela Damjanović. Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions. Journal of Modern Dynamics, 2007, 1 (4) : 665-688. doi: 10.3934/jmd.2007.1.665

[18]

El Houcein El Abdalaoui, Joanna Kułaga-Przymus, Mariusz Lemańczyk, Thierry de la Rue. The Chowla and the Sarnak conjectures from ergodic theory point of view. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2899-2944. doi: 10.3934/dcds.2017125

[19]

A. Gasull, Víctor Mañosa, Xavier Xarles. Rational periodic sequences for the Lyness recurrence. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 587-604. doi: 10.3934/dcds.2012.32.587

[20]

Jean-Paul Thouvenot. The work of Lewis Bowen on the entropy theory of non-amenable group actions. Journal of Modern Dynamics, 2019, 15: 133-141. doi: 10.3934/jmd.2019016

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (193)
  • HTML views (181)
  • Cited by (0)

Other articles
by authors

[Back to Top]