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 & 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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[1]

Felipe García-Ramos, Brian Marcus. Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 729-746. doi: 10.3934/dcds.2019030

[2]

Jie Li. How chaotic is an almost mean equicontinuous system?. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4727-4744. doi: 10.3934/dcds.2018208

[3]

Michel Coornaert, Fabrice Krieger. Mean topological dimension for actions of discrete amenable groups. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 779-793. doi: 10.3934/dcds.2005.13.779

[4]

Hongyong Deng, Wei Wei. Existence and stability analysis for nonlinear optimal control problems with $1$-mean equicontinuous controls. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1409-1422. doi: 10.3934/jimo.2015.11.1409

[5]

Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113

[6]

Dandan Cheng, Qian Hao, Zhiming Li. Scale pressure for amenable group actions. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1091-1102. doi: 10.3934/cpaa.2021008

[7]

Jie Li, Xiangdong Ye, Tao Yu. Mean equicontinuity, complexity and applications. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 359-393. doi: 10.3934/dcds.2020167

[8]

Alexander Vladimirov. Equicontinuous sweeping processes. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 565-573. doi: 10.3934/dcdsb.2013.18.565

[9]

Yoshikazu Katayama, Colin E. Sutherland and Masamichi Takesaki. The intrinsic invariant of an approximately finite dimensional factor and the cocycle conjugacy of discrete amenable group actions. Electronic Research Announcements, 1995, 1: 43-47.

[10]

Xiankun Ren. Periodic measures are dense in invariant measures for residually finite amenable group actions with specification. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1657-1667. doi: 10.3934/dcds.2018068

[11]

Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195

[12]

Fausto Ferrari, Qing Liu, Juan Manfredi. On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2779-2793. doi: 10.3934/dcds.2014.34.2779

[13]

Xiaojun Huang, Zhiqiang Li, Yunhua Zhou. A variational principle of topological pressure on subsets for amenable group actions. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2687-2703. doi: 10.3934/dcds.2020146

[14]

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

[15]

Gerhard Keller. Maximal equicontinuous generic factors and weak model sets. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6855-6875. doi: 10.3934/dcds.2020132

[16]

Thai Son Doan, Martin Rasmussen, Peter E. Kloeden. The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 875-887. doi: 10.3934/dcdsb.2015.20.875

[17]

Dariusz Skrenty. Absolutely continuous spectrum of some group extensions of Gaussian actions. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 365-378. doi: 10.3934/dcds.2010.26.365

[18]

Marko Kostić. Almost periodic type functions and densities. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021008

[19]

Josu Doncel, Nicolas Gast, Bruno Gaujal. Discrete mean field games: Existence of equilibria and convergence. Journal of Dynamics & Games, 2019, 6 (3) : 221-239. doi: 10.3934/jdg.2019016

[20]

Hailong Zhu, Jifeng Chu, Weinian Zhang. Mean-square almost automorphic solutions for stochastic differential equations with hyperbolicity. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1935-1953. doi: 10.3934/dcds.2018078

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (94)
  • HTML views (181)
  • Cited by (0)

Other articles
by authors

[Back to Top]