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Planar S-systems: Global stability and the center problem
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 |
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 [
References:
[1] |
E. Akin, J. Auslander and K. Berg,
When is a transitive map chaotic?, Ohio State Univ. Math. Res. Inst. Publ., 5 (1996), 25-40.
|
[2] |
J. Auslander,
Mean- l-stable systems, Illinois Journal of Mathematics, 3 (1959), 566-579.
|
[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. |
[4] |
M. Baake and U. Grimm, Aperiodic Order, Cambridge University Press, 2013.
doi: 10.1017/CBO9781139025256. |
[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. |
[6] |
M. Baake, D. 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. |
[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. |
[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. |
[9] |
S. Dworkin,
Spectral theory and x-ray diffraction, Journal of mathematical physics, 34 (1993), 2965-2967.
doi: 10.1063/1.530108. |
[10] |
R. Ellis,
Equicontinuity and almost periodic functions, Proceedings of the American Mathematical Society, 10 (1959), 637-643.
doi: 10.2307/2033667. |
[11] |
S. Fomin,
On dynamical systems with pure point spectrum (russian), Dokl. Akad. Nauk SSSR, 77 (1951), 29-32.
|
[12] |
H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981. |
[13] |
H. Furstenberg, Y. 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. |
[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. |
[15] |
R. H. Gilman,
Classes of linear automata, Ergodic Theory Dyn. Syst., 7 (1987), 105-118.
doi: 10.1017/S0143385700003837. |
[16] |
E. Glasner, Ergodic Theory Via Joinings, American Mathematical Society, Providence, RI,
2003.
doi: 10.1090/surv/101. |
[17] |
J.-B. Gouere,
Quasicrystals and almost periodicity, Communications in Mathematical Physics, 255 (2005), 655-681.
doi: 10.1007/s00220-004-1271-8. |
[18] |
W. Huang, P. Lu and X. Ye,
Measure-theoretical sensitivity and equicontinuity, Israel Journal of Mathematics, 183 (2011), 233-283.
doi: 10.1007/s11856-011-0049-x. |
[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. |
[20] |
J.-Y. Lee, R. 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. |
[21] |
D. Lenz, An autocorrelation and discrete spectrum for dynamical systems on metric spaces,
arXiv preprint, arXiv: 1608.05636. |
[22] |
J. Li, S. Tu and X. Ye,
Mean equicontinuity and mean sensitivity, Ergodic Theory and Dynamical Systems, 35 (2015), 2587-2612.
doi: 10.1017/etds.2014.41. |
[23] |
E. Lindenstrauss,
Pointwise theorems for amenable groups, Inventiones mathematicae, 146 (2001), 259-295.
doi: 10.1007/s002220100162. |
[24] |
J. C. Oxtoby,
Ergodic sets, Bulletin of the American Mathematical Society, 58 (1952), 116-136.
doi: 10.1090/S0002-9904-1952-09580-X. |
[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. |
[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. |
[27] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. SpringerVerlag, New York-Berlin, 1982. |
show all references
References:
[1] |
E. Akin, J. Auslander and K. Berg,
When is a transitive map chaotic?, Ohio State Univ. Math. Res. Inst. Publ., 5 (1996), 25-40.
|
[2] |
J. Auslander,
Mean- l-stable systems, Illinois Journal of Mathematics, 3 (1959), 566-579.
|
[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. |
[4] |
M. Baake and U. Grimm, Aperiodic Order, Cambridge University Press, 2013.
doi: 10.1017/CBO9781139025256. |
[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. |
[6] |
M. Baake, D. 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. |
[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. |
[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. |
[9] |
S. Dworkin,
Spectral theory and x-ray diffraction, Journal of mathematical physics, 34 (1993), 2965-2967.
doi: 10.1063/1.530108. |
[10] |
R. Ellis,
Equicontinuity and almost periodic functions, Proceedings of the American Mathematical Society, 10 (1959), 637-643.
doi: 10.2307/2033667. |
[11] |
S. Fomin,
On dynamical systems with pure point spectrum (russian), Dokl. Akad. Nauk SSSR, 77 (1951), 29-32.
|
[12] |
H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981. |
[13] |
H. Furstenberg, Y. 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. |
[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. |
[15] |
R. H. Gilman,
Classes of linear automata, Ergodic Theory Dyn. Syst., 7 (1987), 105-118.
doi: 10.1017/S0143385700003837. |
[16] |
E. Glasner, Ergodic Theory Via Joinings, American Mathematical Society, Providence, RI,
2003.
doi: 10.1090/surv/101. |
[17] |
J.-B. Gouere,
Quasicrystals and almost periodicity, Communications in Mathematical Physics, 255 (2005), 655-681.
doi: 10.1007/s00220-004-1271-8. |
[18] |
W. Huang, P. Lu and X. Ye,
Measure-theoretical sensitivity and equicontinuity, Israel Journal of Mathematics, 183 (2011), 233-283.
doi: 10.1007/s11856-011-0049-x. |
[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. |
[20] |
J.-Y. Lee, R. 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. |
[21] |
D. Lenz, An autocorrelation and discrete spectrum for dynamical systems on metric spaces,
arXiv preprint, arXiv: 1608.05636. |
[22] |
J. Li, S. Tu and X. Ye,
Mean equicontinuity and mean sensitivity, Ergodic Theory and Dynamical Systems, 35 (2015), 2587-2612.
doi: 10.1017/etds.2014.41. |
[23] |
E. Lindenstrauss,
Pointwise theorems for amenable groups, Inventiones mathematicae, 146 (2001), 259-295.
doi: 10.1007/s002220100162. |
[24] |
J. C. Oxtoby,
Ergodic sets, Bulletin of the American Mathematical Society, 58 (1952), 116-136.
doi: 10.1090/S0002-9904-1952-09580-X. |
[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. |
[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. |
[27] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. SpringerVerlag, New York-Berlin, 1982. |
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