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August 2017, 37(8): 4543-4563. doi: 10.3934/dcds.2017194

Measurable sensitivity via Furstenberg families

Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China

* Corresponding author: Tao Yu

Received  September 2016 Revised  March 2017 Published  April 2017

Fund Project: The author was supported by NNSF of China (11371339,11431012,11571335)

Let $(X, T)$ be a topological dynamical system, and $\mu$ be a $T$-invariant Borel probability measure on $X$. Let $\mathcal{F}$ be a family of subsets of $\mathbb{Z}_+$. We introduce notions of $\mathcal{F}$-sensitivity for $\mu$ and block $\mathcal{F}$-sensitivity for $\mu$.

Let $\mathcal{F}_t$ (resp. $\mathcal{F}_{ip}$) be the families consisting of thick sets (resp. IP-sets). The following Auslander-Yorke's type dichotomy theorems are obtained: (1) a minimal system is either $\mathcal{F}_{t}$-sensitive for $\mu$ or an almost one-to-one extension of its maximal equicontinous factor. (2) a minimal system is either block $\mathcal{F}_{t}$-sensitive for $\mu$ or a proximal extension of its maximal equicontinous factor. (3) a minimal system is either block $\mathcal{F}_{ip}$-sensitive for $\mu$ or an almost one-to-one extension of its $\infty$-step nilfactor.

We also introduce the notion of topological $l$-sensitivity, and construct a minimal system which is $l$-sensitive but not $(l+1)$-sensitive for $l\in\mathbb{N}$.

Citation: Tao Yu. Measurable sensitivity via Furstenberg families. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4543-4563. doi: 10.3934/dcds.2017194
References:
[1]

C. AbrahamG. Biau and B. Cadre, Chaotic properties of mappings on a probability space, J. Math. Anal. Appl., 266 (2002), 420-431. doi: 10.1006/jmaa.2001.7754.

[2]

E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433. doi: 10.1088/0951-7715/16/4/313.

[3]

J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies 153, Elsevier, 1988.

[4]

J. Auslander and J. A. Yorke, Interval maps, factors of maps, and chaos, Tôhoku Math. J. (2), 32 (1980), 177-188. doi: 10.2748/tmj/1178229634.

[5]

V. Bergelson, Ultrafilters, IP sets, dynamics, and combinatorial number theory, Ultrafilters Across Mathematics, 23-47, Contemp. Math. , 530, Amer. Math. Soc. , Providence, RI, 2010. doi: 10.1090/conm/530/10439.

[6]

B. Cadre and P. Jacob, On pairwise sensitivity, J. Math. Anal. Appl., 309 (2005), 375-382. doi: 10.1016/j.jmaa.2005.01.061.

[7]

P. DongS. DonosoA. MaassS. Shao and X. Ye, Infinite-step nilsystems, independence and complexity, Ergodic Theory Dynam. Systems, 33 (2013), 118-143. doi: 10.1017/S0143385711000861.

[8]

T. Downarowicz and E. Glasner, Isomorphic extension and applications, Topol. Methods Nonlinear Anal., 48 (2016), 321-338. doi: 10.12775/TMNA.2016.050.

[9]

T. Downarowicz and S. Kasjan, Odometers and toeplitz systems revisited in the context of sarnak's conjecture, Studia Math., 229 (2015), 45-72.

[10]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures. Princeton University Press, Princeton, N. J. , 1981.

[11]

H. Furstenberg and B. Weiss, On almost 1-1 extensions, Israel J. Math., 65 (1989), 311-322. doi: 10.1007/BF02764869.

[12]

F. García-Ramos, Weak Forms of Topological and Measure Theoretical Equicontinuity: Relationships with Discrete Spectrum and Sequence Entropy, Ergodic Theory Dynam. Systems, to appear.

[13]

J. Gillis, Notes on a property of measurable sets, J. London Math. Soc., 11 (1936), 139-141. doi: 10.1112/jlms/s1-11.2.139.

[14]

E. Glasner, Y. Gutman and X. Ye, Higher order regionally proximal equivalence relations for general group actions, preprint.

[15]

I. GrigorievM. C. IordanA. LubinN. Ince and C. E. Silva, On μ-compatible metrics and measurable sensitivity, Colloq. Math., 126 (2012), 53-72. doi: 10.4064/cm126-1-3.

[16]

J. HallettL. Manuelli and C. E. Silva, On Li-Yorke measurable sensitivity, Proc. Amer. Math. Soc., 143 (2015), 2411-2426. doi: 10.1090/S0002-9939-2015-12430-6.

[17]

B. HostB. Kra and A. Maass, Nilsequences and a structure theory for topological dynamical systems, Adv. Math., 224 (2010), 103-129. doi: 10.1016/j.aim.2009.11.009.

[18]

W. HuangD. KhilkoS. Kolyada and G. H. Zhang, Dynamical compactness and sensitivity, J. Differential Equations, 260 (2016), 6800-6827. doi: 10.1016/j.jde.2016.01.011.

[19]

W. Huang, S. Kolyada and G. H. Zhang, Analogues of Auslander-Yorke theorems for multisensitivity, Ergodic Theory Dynam. Systems, to appear. doi: 10.1017/etds.2016.48.

[20]

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

[21]

W. HuangS. Shao and X. Ye, Nil Bohr-sets and almost automorphy of higher order, Mem. Amer. Math. Soc., 241 (2016), v+83 pp. doi: 10.1090/memo/1143.

[22]

W. Huang and X. Ye, Topological complexity, return times and weak disjointness, Ergodic Theory Dynam. Systems, 24 (2004), 825-846. doi: 10.1017/S0143385703000543.

[23]

J. JamesT. KoberdaK. LindseyC. E. Silva and P. Speh, Measurable sensitivity, Proc. Amer. Math. Soc., 136 (2008), 3549-3559. doi: 10.1090/S0002-9939-08-09294-0.

[24]

J. Li, Dynamical characterization of C-sets and its application, Fund. Math., 216 (2012), 259-286. doi: 10.4064/fm216-3-4.

[25]

J. Li, Measure-theoretic sensitivity via finite partitions, Nonlinearity, 29 (2016), 2133-2144. doi: 10.1088/0951-7715/29/7/2133.

[26]

J. Li and X. Ye, Recent development of chaos theory in topological dynamics, Acta Math. Sin. (Engl. Ser.), 32 (2016), 83-114. doi: 10.1007/s10114-015-4574-0.

[27]

R. Li and Y. Shi, Stronger forms of sensitivity for measure-preserving maps and semiflows on probability spaces, Abstr. Appl. Anal. , Art. , (2014), ID 769523, 10 pages. doi: 10.1155/2014/769523.

[28]

H. Liu, L. Liao and L. Wang, Thickly syndetical sensitivity of topological dynamical system, Discrete Dyn. Nat. Soc. , Art. , (2014), ID 583431, 4 pages. doi: 10.1155/2014/583431.

[29]

T. K. S. Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity, 20 (2007), 2115-2126. doi: 10.1088/0951-7715/20/9/006.

[30]

D. Ruelle, Dynamical systems with turbulent behavior, Mathematical problems in theoretical physics (Proc. Internat. Conf. , Univ. Rome, Rome, 1977), Lecture Notes in Phys. , vol. 80, Springer, Berlin-New York, 1978, pp. 341-360.

[31]

S. Shao and X. Ye, Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence, Adv. Math., 231 (2012), 1786-1817. doi: 10.1016/j.aim.2012.07.012.

[32]

H. Wu and H. Wang, Measure-theoretical sensitivity and scrambled sets via Furstenberg families, J. Dyn. Syst. Geom. Theor., 7 (2009), 1-12. doi: 10.1080/1726037X.2009.10698558.

[33]

X. Ye and T. Yu, Sensitivity, proximal extension and higher order almost automorphy, Trans. Amer. Math. Soc., 13 (2001). doi: 10.1515/form.2001.023.

[34]

X. Ye and R. Zhang, On sensitive sets in topological dynamics, Nonlinearity, 21 (2008), 1601-1620. doi: 10.1088/0951-7715/21/7/012.

[35]

R. Zhang, On sensitivity, Sequence Entropy and Related Problems in Dynamical Systems, Ph. D thesis, University of Science and Technology of China, 2008.

show all references

References:
[1]

C. AbrahamG. Biau and B. Cadre, Chaotic properties of mappings on a probability space, J. Math. Anal. Appl., 266 (2002), 420-431. doi: 10.1006/jmaa.2001.7754.

[2]

E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421-1433. doi: 10.1088/0951-7715/16/4/313.

[3]

J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies 153, Elsevier, 1988.

[4]

J. Auslander and J. A. Yorke, Interval maps, factors of maps, and chaos, Tôhoku Math. J. (2), 32 (1980), 177-188. doi: 10.2748/tmj/1178229634.

[5]

V. Bergelson, Ultrafilters, IP sets, dynamics, and combinatorial number theory, Ultrafilters Across Mathematics, 23-47, Contemp. Math. , 530, Amer. Math. Soc. , Providence, RI, 2010. doi: 10.1090/conm/530/10439.

[6]

B. Cadre and P. Jacob, On pairwise sensitivity, J. Math. Anal. Appl., 309 (2005), 375-382. doi: 10.1016/j.jmaa.2005.01.061.

[7]

P. DongS. DonosoA. MaassS. Shao and X. Ye, Infinite-step nilsystems, independence and complexity, Ergodic Theory Dynam. Systems, 33 (2013), 118-143. doi: 10.1017/S0143385711000861.

[8]

T. Downarowicz and E. Glasner, Isomorphic extension and applications, Topol. Methods Nonlinear Anal., 48 (2016), 321-338. doi: 10.12775/TMNA.2016.050.

[9]

T. Downarowicz and S. Kasjan, Odometers and toeplitz systems revisited in the context of sarnak's conjecture, Studia Math., 229 (2015), 45-72.

[10]

H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures. Princeton University Press, Princeton, N. J. , 1981.

[11]

H. Furstenberg and B. Weiss, On almost 1-1 extensions, Israel J. Math., 65 (1989), 311-322. doi: 10.1007/BF02764869.

[12]

F. García-Ramos, Weak Forms of Topological and Measure Theoretical Equicontinuity: Relationships with Discrete Spectrum and Sequence Entropy, Ergodic Theory Dynam. Systems, to appear.

[13]

J. Gillis, Notes on a property of measurable sets, J. London Math. Soc., 11 (1936), 139-141. doi: 10.1112/jlms/s1-11.2.139.

[14]

E. Glasner, Y. Gutman and X. Ye, Higher order regionally proximal equivalence relations for general group actions, preprint.

[15]

I. GrigorievM. C. IordanA. LubinN. Ince and C. E. Silva, On μ-compatible metrics and measurable sensitivity, Colloq. Math., 126 (2012), 53-72. doi: 10.4064/cm126-1-3.

[16]

J. HallettL. Manuelli and C. E. Silva, On Li-Yorke measurable sensitivity, Proc. Amer. Math. Soc., 143 (2015), 2411-2426. doi: 10.1090/S0002-9939-2015-12430-6.

[17]

B. HostB. Kra and A. Maass, Nilsequences and a structure theory for topological dynamical systems, Adv. Math., 224 (2010), 103-129. doi: 10.1016/j.aim.2009.11.009.

[18]

W. HuangD. KhilkoS. Kolyada and G. H. Zhang, Dynamical compactness and sensitivity, J. Differential Equations, 260 (2016), 6800-6827. doi: 10.1016/j.jde.2016.01.011.

[19]

W. Huang, S. Kolyada and G. H. Zhang, Analogues of Auslander-Yorke theorems for multisensitivity, Ergodic Theory Dynam. Systems, to appear. doi: 10.1017/etds.2016.48.

[20]

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

[21]

W. HuangS. Shao and X. Ye, Nil Bohr-sets and almost automorphy of higher order, Mem. Amer. Math. Soc., 241 (2016), v+83 pp. doi: 10.1090/memo/1143.

[22]

W. Huang and X. Ye, Topological complexity, return times and weak disjointness, Ergodic Theory Dynam. Systems, 24 (2004), 825-846. doi: 10.1017/S0143385703000543.

[23]

J. JamesT. KoberdaK. LindseyC. E. Silva and P. Speh, Measurable sensitivity, Proc. Amer. Math. Soc., 136 (2008), 3549-3559. doi: 10.1090/S0002-9939-08-09294-0.

[24]

J. Li, Dynamical characterization of C-sets and its application, Fund. Math., 216 (2012), 259-286. doi: 10.4064/fm216-3-4.

[25]

J. Li, Measure-theoretic sensitivity via finite partitions, Nonlinearity, 29 (2016), 2133-2144. doi: 10.1088/0951-7715/29/7/2133.

[26]

J. Li and X. Ye, Recent development of chaos theory in topological dynamics, Acta Math. Sin. (Engl. Ser.), 32 (2016), 83-114. doi: 10.1007/s10114-015-4574-0.

[27]

R. Li and Y. Shi, Stronger forms of sensitivity for measure-preserving maps and semiflows on probability spaces, Abstr. Appl. Anal. , Art. , (2014), ID 769523, 10 pages. doi: 10.1155/2014/769523.

[28]

H. Liu, L. Liao and L. Wang, Thickly syndetical sensitivity of topological dynamical system, Discrete Dyn. Nat. Soc. , Art. , (2014), ID 583431, 4 pages. doi: 10.1155/2014/583431.

[29]

T. K. S. Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity, 20 (2007), 2115-2126. doi: 10.1088/0951-7715/20/9/006.

[30]

D. Ruelle, Dynamical systems with turbulent behavior, Mathematical problems in theoretical physics (Proc. Internat. Conf. , Univ. Rome, Rome, 1977), Lecture Notes in Phys. , vol. 80, Springer, Berlin-New York, 1978, pp. 341-360.

[31]

S. Shao and X. Ye, Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence, Adv. Math., 231 (2012), 1786-1817. doi: 10.1016/j.aim.2012.07.012.

[32]

H. Wu and H. Wang, Measure-theoretical sensitivity and scrambled sets via Furstenberg families, J. Dyn. Syst. Geom. Theor., 7 (2009), 1-12. doi: 10.1080/1726037X.2009.10698558.

[33]

X. Ye and T. Yu, Sensitivity, proximal extension and higher order almost automorphy, Trans. Amer. Math. Soc., 13 (2001). doi: 10.1515/form.2001.023.

[34]

X. Ye and R. Zhang, On sensitive sets in topological dynamics, Nonlinearity, 21 (2008), 1601-1620. doi: 10.1088/0951-7715/21/7/012.

[35]

R. Zhang, On sensitivity, Sequence Entropy and Related Problems in Dynamical Systems, Ph. D thesis, University of Science and Technology of China, 2008.

Figure 1.  $B^{(1)}, B^{(2)}$
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