December  2021, 41(12): 5871-5886. doi: 10.3934/dcds.2021099

Discrete spectrum for amenable group actions

1. 

Department of Mathematics, Shantou University, Shantou 515063, Guangdong, China

2. 

School of Mathematical Sciences and Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200433, China

3. 

School of Mathematics, Hefei University of Technology, Hefei 230009, Anhui, China

* Corresponding author: Guohua Zhang

Received  December 2020 Revised  May 2021 Published  December 2021 Early access  July 2021

In this paper, we study discrete spectrum of invariant measures for countable discrete amenable group actions.

We show that an invariant measure has discrete spectrum if and only if it has bounded measure complexity. We also prove that, discrete spectrum can be characterized via measure-theoretic complexity using names of a partition and the Hamming distance, and it turns out to be equivalent to both mean equicontinuity and equicontinuity in the mean.

Citation: Tao Yu, Guohua Zhang, Ruifeng Zhang. Discrete spectrum for amenable group actions. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5871-5886. doi: 10.3934/dcds.2021099
References:
[1]

S. Ferenczi, Measure-theoretic complexity of ergodic systems, Israel J. Math., 100 (1997), 189-207.  doi: 10.1007/BF02773640.

[2]

E. Følner, On groups with full {B}anach mean value, Math. Scand., 3 (1955), 243-254.  doi: 10.7146/math.scand.a-10442.

[3]

G. Fuhrmann, M. Gröger and D. Lenz, The structure of mean equicontinuous group actions, preprint, arXiv: 1812.10219v1.

[4]

F. García-Ramos, Weak forms of topological and measure-theoretical equicontinuity: Relationships with discrete spectrum and sequence entropy, Ergodic Theory Dynam. Systems, 37 (2017), 1211-1237.  doi: 10.1017/etds.2015.83.

[5]

F. García-Ramos and B. Marcus, Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems, Discrete Contin. Dyn. Syst., 39 (2019), 729-746.  doi: 10.3934/dcds.2019030.

[6]

P. R. Halmos and J. von Neumann, Operator methods in classical mechanics. II, Ann. of Math. (2), 43 (1942), 332-350.  doi: 10.2307/1968872.

[7]

W. HuangJ. LiJ.-P. ThouvenotL. Xu and X. Ye, Bounded complexity, mean equicontinuity and discrete spectrum, Ergodic Theory Dynam. Systems, 41 (2021), 494-533.  doi: 10.1017/etds.2019.66.

[8]

W. HuangZ. Wang and X. Ye, Measure complexity and Möbius disjointness, Adv. Math., 347 (2019), 827-858.  doi: 10.1016/j.aim.2019.03.007.

[9]

A. B. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137–173.

[10]

A. B. Katok and J.-P. Thouvenot, Slow entropy type invariants and smooth realization of commuting measure-preserving transformations, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 323-338.  doi: 10.1016/S0246-0203(97)80094-5.

[11]

D. Lenz, An autocorrelation and discrete spectrum for dynamical systems on metric spaces, Ergodic Theory Dynam. Systems, 41 (2021), 906-922.  doi: 10.1017/etds.2019.102.

[12]

D. Lenz and N. Strungaru, Pure point spectrum for measure dynamical systems on locally compact abelian groups, J. Math. Pures Appl., 92 (2009), no. 4, 323–341. doi: 10.1016/j. matpur. 2009.05.013.

[13]

E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.  doi: 10.1007/s002220100162.

[14]

J. M. Ollagnier, Ergodic theory and statistical mechanics, Lecture Notes in Mathematics, 1115, Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0101575.

[15]

D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987), 1-141.  doi: 10.1007/BF02790325.

[16]

K. Sakai, On compact transformation semigroups with discrete spectrum, Sci. Rep. Kagoshima Univ., 34 (1985), 11-17. 

[17]

B. Scarpellini, Stability properties of flows with pure point spectrum, J. London Math. Soc. (2), 26 (1982), 451-464.  doi: 10.1112/jlms/s2-26.3.451.

[18]

A. M. Vershik, Dynamics of metrics in measure spaces and their asymptotic invariants, Markov Process. Related Fields, 16 (2010), 169-184. 

[19]

A. M. Vershik, Scaled entropy and automorphisms with a pure point spectrum, Algebra i Analiz, 23 (2011), 111-135.  doi: 10.1090/S1061-0022-2011-01187-2.

[20]

A. M. VershikP. B. Zatitskiy and F. V. Petrov, Geometry and dynamics of admissible metrics in measure spaces, Cent. Eur. J. Math., 11 (2013), 379-400.  doi: 10.2478/s11533-012-0149-9.

[21]

P. Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[22]

T. B. Ward and Q. Zhang, The {A}bramov-{R}okhlin entropy addition formula for amenable group actions, Monatsh. Math., 114 (1992), 317-329.  doi: 10.1007/BF01299386.

[23]

T. Yu, Measure-theoretic mean equicontinuity and bounded complexity, J. Differential Equations, 267 (2019), 6152-6170.  doi: 10.1016/j.jde.2019.06.017.

[24]

R. J. Zimmer, Ergodic actions with generalized discrete spectrum, Illinois J. Math., 20 (1976), 555-588. 

show all references

References:
[1]

S. Ferenczi, Measure-theoretic complexity of ergodic systems, Israel J. Math., 100 (1997), 189-207.  doi: 10.1007/BF02773640.

[2]

E. Følner, On groups with full {B}anach mean value, Math. Scand., 3 (1955), 243-254.  doi: 10.7146/math.scand.a-10442.

[3]

G. Fuhrmann, M. Gröger and D. Lenz, The structure of mean equicontinuous group actions, preprint, arXiv: 1812.10219v1.

[4]

F. García-Ramos, Weak forms of topological and measure-theoretical equicontinuity: Relationships with discrete spectrum and sequence entropy, Ergodic Theory Dynam. Systems, 37 (2017), 1211-1237.  doi: 10.1017/etds.2015.83.

[5]

F. García-Ramos and B. Marcus, Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems, Discrete Contin. Dyn. Syst., 39 (2019), 729-746.  doi: 10.3934/dcds.2019030.

[6]

P. R. Halmos and J. von Neumann, Operator methods in classical mechanics. II, Ann. of Math. (2), 43 (1942), 332-350.  doi: 10.2307/1968872.

[7]

W. HuangJ. LiJ.-P. ThouvenotL. Xu and X. Ye, Bounded complexity, mean equicontinuity and discrete spectrum, Ergodic Theory Dynam. Systems, 41 (2021), 494-533.  doi: 10.1017/etds.2019.66.

[8]

W. HuangZ. Wang and X. Ye, Measure complexity and Möbius disjointness, Adv. Math., 347 (2019), 827-858.  doi: 10.1016/j.aim.2019.03.007.

[9]

A. B. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137–173.

[10]

A. B. Katok and J.-P. Thouvenot, Slow entropy type invariants and smooth realization of commuting measure-preserving transformations, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 323-338.  doi: 10.1016/S0246-0203(97)80094-5.

[11]

D. Lenz, An autocorrelation and discrete spectrum for dynamical systems on metric spaces, Ergodic Theory Dynam. Systems, 41 (2021), 906-922.  doi: 10.1017/etds.2019.102.

[12]

D. Lenz and N. Strungaru, Pure point spectrum for measure dynamical systems on locally compact abelian groups, J. Math. Pures Appl., 92 (2009), no. 4, 323–341. doi: 10.1016/j. matpur. 2009.05.013.

[13]

E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.  doi: 10.1007/s002220100162.

[14]

J. M. Ollagnier, Ergodic theory and statistical mechanics, Lecture Notes in Mathematics, 1115, Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0101575.

[15]

D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987), 1-141.  doi: 10.1007/BF02790325.

[16]

K. Sakai, On compact transformation semigroups with discrete spectrum, Sci. Rep. Kagoshima Univ., 34 (1985), 11-17. 

[17]

B. Scarpellini, Stability properties of flows with pure point spectrum, J. London Math. Soc. (2), 26 (1982), 451-464.  doi: 10.1112/jlms/s2-26.3.451.

[18]

A. M. Vershik, Dynamics of metrics in measure spaces and their asymptotic invariants, Markov Process. Related Fields, 16 (2010), 169-184. 

[19]

A. M. Vershik, Scaled entropy and automorphisms with a pure point spectrum, Algebra i Analiz, 23 (2011), 111-135.  doi: 10.1090/S1061-0022-2011-01187-2.

[20]

A. M. VershikP. B. Zatitskiy and F. V. Petrov, Geometry and dynamics of admissible metrics in measure spaces, Cent. Eur. J. Math., 11 (2013), 379-400.  doi: 10.2478/s11533-012-0149-9.

[21]

P. Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[22]

T. B. Ward and Q. Zhang, The {A}bramov-{R}okhlin entropy addition formula for amenable group actions, Monatsh. Math., 114 (1992), 317-329.  doi: 10.1007/BF01299386.

[23]

T. Yu, Measure-theoretic mean equicontinuity and bounded complexity, J. Differential Equations, 267 (2019), 6152-6170.  doi: 10.1016/j.jde.2019.06.017.

[24]

R. J. Zimmer, Ergodic actions with generalized discrete spectrum, Illinois J. Math., 20 (1976), 555-588. 

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