February  2019, 39(2): 729-746. doi: 10.3934/dcds.2019030

Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems

1. 

CONACyT / Instituto de Física, Universidad Autónoma de San Luis Potosí (UASLP), Av. Manuel Nava #6, Zona Universitaria, San Luis Potosí, S.L.P., 78290, México

2. 

Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

* Corresponding author

Received  September 2017 Revised  August 2018 Published  November 2018

We show that a continuous abelian action (in particular $\mathbb{R}^{d}$) on a compact metric space equipped with an invariant ergodic measure has discrete spectrum if and only it is $μ-$mean equicontinuous (proven for $\mathbb{Z}^{d}$ in [14]). In order to do this we introduce mean equicontinuity and mean sensitivity with respect to a function. We study this notion in the topological and measure theoretic setting. In the measure theoretic case we characterize almost periodic functions with these concepts and in the topological case we show that weakly almost periodic functions are mean equicontinuous (the converse does not hold). We compare our results with some results in the theory of Delone dynamical systems and quasicrystals.

Citation: Felipe García-Ramos, Brian Marcus. Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 729-746. doi: 10.3934/dcds.2019030
References:
[1]

E. AkinJ. Auslander and K. Berg, When is a transitive map chaotic?, Ohio State Univ. Math. Res. Inst. Publ., 5 (1996), 25-40.   Google Scholar

[2]

J. Auslander, Mean- l-stable systems, Illinois Journal of Mathematics, 3 (1959), 566-579.   Google Scholar

[3]

J. Auslander and J. A. Yorke, Interval maps, factors of maps, and chaos, Tohoku Mathematical Journal, 32 (1980), 177-188.  doi: 10.2748/tmj/1178229634.  Google Scholar

[4]

M. Baake and U. Grimm, Aperiodic Order, Cambridge University Press, 2013. doi: 10.1017/CBO9781139025256.  Google Scholar

[5]

M. Baake and D. Lenz, Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra, Ergodic Theory and Dynamical Systems, 24 (2004), 1867-1893.  doi: 10.1017/S0143385704000318.  Google Scholar

[6]

M. BaakeD. Lenz and R. V. Moody, Characterization of model sets by dynamical systems, Ergodic Theory and Dynamical Systems, 27 (2007), 341-382.  doi: 10.1017/S0143385706000800.  Google Scholar

[7]

B. Cadre and P. Jacob, On pairwise sensitivity, Journal of Mathematical Analysis and Applications, 309 (2005), 375-382.  doi: 10.1016/j.jmaa.2005.01.061.  Google Scholar

[8]

T. Downarowicz and E. Glasner, Isomorphic extensions and applications, Topological Methods in Nonlinear Analysis, 48 (2016), 321-338.  doi: 10.12775/TMNA.2016.050.  Google Scholar

[9]

S. Dworkin, Spectral theory and x-ray diffraction, Journal of mathematical physics, 34 (1993), 2965-2967.  doi: 10.1063/1.530108.  Google Scholar

[10]

R. Ellis, Equicontinuity and almost periodic functions, Proceedings of the American Mathematical Society, 10 (1959), 637-643.  doi: 10.2307/2033667.  Google Scholar

[11]

S. Fomin, On dynamical systems with pure point spectrum (russian), Dokl. Akad. Nauk SSSR, 77 (1951), 29-32.   Google Scholar

[12]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981.  Google Scholar

[13]

H. FurstenbergY. Katznelson and D. Ornstein, The ergodic theoretical proof of szemerédi's theorem, Bulletin of the AMS, 7 (1982), 527-552.  doi: 10.1090/S0273-0979-1982-15052-2.  Google Scholar

[14]

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

[15]

R. H. Gilman, Classes of linear automata, Ergodic Theory Dyn. Syst., 7 (1987), 105-118.  doi: 10.1017/S0143385700003837.  Google Scholar

[16]

E. Glasner, Ergodic Theory Via Joinings, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar

[17]

J.-B. Gouere, Quasicrystals and almost periodicity, Communications in Mathematical Physics, 255 (2005), 655-681.  doi: 10.1007/s00220-004-1271-8.  Google Scholar

[18]

W. HuangP. Lu and X. Ye, Measure-theoretical sensitivity and equicontinuity, Israel Journal of Mathematics, 183 (2011), 233-283.  doi: 10.1007/s11856-011-0049-x.  Google Scholar

[19]

J. Kellendonk, D. Lenz and J. Savinien, Mathematics of Aperiodic Order, Progress in Mathematics, 309. Birkh?user/Springer, Basel, 2015. doi: 10.1007/978-3-0348-0903-0.  Google Scholar

[20]

J.-Y. LeeR. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra, Annales Henri Poincaré, 3 (2002), 1003-1018.  doi: 10.1007/s00023-002-8646-1.  Google Scholar

[21]

D. Lenz, An autocorrelation and discrete spectrum for dynamical systems on metric spaces, arXiv preprint, arXiv: 1608.05636. Google Scholar

[22]

J. LiS. Tu and X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory and Dynamical Systems, 35 (2015), 2587-2612.  doi: 10.1017/etds.2014.41.  Google Scholar

[23]

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

[24]

J. C. Oxtoby, Ergodic sets, Bulletin of the American Mathematical Society, 58 (1952), 116-136.  doi: 10.1090/S0002-9904-1952-09580-X.  Google Scholar

[25]

E. Robinson Jr, The dynamical properties of penrose tilings, Transactions of the American Mathematical Society, 348 (1996), 4447-4464.  doi: 10.1090/S0002-9947-96-01640-6.  Google Scholar

[26]

B. Scarpellini, Stability properties of flows with pure point spectrum, Journal of the London Mathematical Society, 26 (1982), 451-464.  doi: 10.1112/jlms/s2-26.3.451.  Google Scholar

[27]

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

show all references

References:
[1]

E. AkinJ. Auslander and K. Berg, When is a transitive map chaotic?, Ohio State Univ. Math. Res. Inst. Publ., 5 (1996), 25-40.   Google Scholar

[2]

J. Auslander, Mean- l-stable systems, Illinois Journal of Mathematics, 3 (1959), 566-579.   Google Scholar

[3]

J. Auslander and J. A. Yorke, Interval maps, factors of maps, and chaos, Tohoku Mathematical Journal, 32 (1980), 177-188.  doi: 10.2748/tmj/1178229634.  Google Scholar

[4]

M. Baake and U. Grimm, Aperiodic Order, Cambridge University Press, 2013. doi: 10.1017/CBO9781139025256.  Google Scholar

[5]

M. Baake and D. Lenz, Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra, Ergodic Theory and Dynamical Systems, 24 (2004), 1867-1893.  doi: 10.1017/S0143385704000318.  Google Scholar

[6]

M. BaakeD. Lenz and R. V. Moody, Characterization of model sets by dynamical systems, Ergodic Theory and Dynamical Systems, 27 (2007), 341-382.  doi: 10.1017/S0143385706000800.  Google Scholar

[7]

B. Cadre and P. Jacob, On pairwise sensitivity, Journal of Mathematical Analysis and Applications, 309 (2005), 375-382.  doi: 10.1016/j.jmaa.2005.01.061.  Google Scholar

[8]

T. Downarowicz and E. Glasner, Isomorphic extensions and applications, Topological Methods in Nonlinear Analysis, 48 (2016), 321-338.  doi: 10.12775/TMNA.2016.050.  Google Scholar

[9]

S. Dworkin, Spectral theory and x-ray diffraction, Journal of mathematical physics, 34 (1993), 2965-2967.  doi: 10.1063/1.530108.  Google Scholar

[10]

R. Ellis, Equicontinuity and almost periodic functions, Proceedings of the American Mathematical Society, 10 (1959), 637-643.  doi: 10.2307/2033667.  Google Scholar

[11]

S. Fomin, On dynamical systems with pure point spectrum (russian), Dokl. Akad. Nauk SSSR, 77 (1951), 29-32.   Google Scholar

[12]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981.  Google Scholar

[13]

H. FurstenbergY. Katznelson and D. Ornstein, The ergodic theoretical proof of szemerédi's theorem, Bulletin of the AMS, 7 (1982), 527-552.  doi: 10.1090/S0273-0979-1982-15052-2.  Google Scholar

[14]

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

[15]

R. H. Gilman, Classes of linear automata, Ergodic Theory Dyn. Syst., 7 (1987), 105-118.  doi: 10.1017/S0143385700003837.  Google Scholar

[16]

E. Glasner, Ergodic Theory Via Joinings, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar

[17]

J.-B. Gouere, Quasicrystals and almost periodicity, Communications in Mathematical Physics, 255 (2005), 655-681.  doi: 10.1007/s00220-004-1271-8.  Google Scholar

[18]

W. HuangP. Lu and X. Ye, Measure-theoretical sensitivity and equicontinuity, Israel Journal of Mathematics, 183 (2011), 233-283.  doi: 10.1007/s11856-011-0049-x.  Google Scholar

[19]

J. Kellendonk, D. Lenz and J. Savinien, Mathematics of Aperiodic Order, Progress in Mathematics, 309. Birkh?user/Springer, Basel, 2015. doi: 10.1007/978-3-0348-0903-0.  Google Scholar

[20]

J.-Y. LeeR. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra, Annales Henri Poincaré, 3 (2002), 1003-1018.  doi: 10.1007/s00023-002-8646-1.  Google Scholar

[21]

D. Lenz, An autocorrelation and discrete spectrum for dynamical systems on metric spaces, arXiv preprint, arXiv: 1608.05636. Google Scholar

[22]

J. LiS. Tu and X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory and Dynamical Systems, 35 (2015), 2587-2612.  doi: 10.1017/etds.2014.41.  Google Scholar

[23]

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

[24]

J. C. Oxtoby, Ergodic sets, Bulletin of the American Mathematical Society, 58 (1952), 116-136.  doi: 10.1090/S0002-9904-1952-09580-X.  Google Scholar

[25]

E. Robinson Jr, The dynamical properties of penrose tilings, Transactions of the American Mathematical Society, 348 (1996), 4447-4464.  doi: 10.1090/S0002-9947-96-01640-6.  Google Scholar

[26]

B. Scarpellini, Stability properties of flows with pure point spectrum, Journal of the London Mathematical Society, 26 (1982), 451-464.  doi: 10.1112/jlms/s2-26.3.451.  Google Scholar

[27]

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

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