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Study on strong sensitivity of systems satisfying the large deviations theorem. (English) Zbl 1479.37007

Summary: Let \((S,\tau)\) be a nontrivial compact metric space with metric \(\tau\) and \(g:S\to S\) be a continuous self-map, \(\mathcal{B}(S)\) be the sigma-algebra of Borel subsets of \(S\), and \(\nu\) be a Borel probability measure on \((S,\mathcal{B}(S))\) with \(\nu(V)\in(0,+\infty)\) for any open subset \(V\neq\emptyset\) of \(S\). This paper proves the following results:
(1)
If the pair \((g,\nu)\) has the property that for any \(\beta>0\), there is \(r(\beta)>0\) such that \[ \nu\biggl(\biggl\{p\in S:\biggl|\frac{1}{m}\sum\limits_{i=0}^{m-1} \mathcal{X}_V (g^i(p))-\int\mathcal{X}_V d\nu\biggr|>\beta\biggr\}\biggr)\leq e^{-mr(\beta)}, \] for any open subset \(V\neq\emptyset\) of \(S\) and all \(n\geq 1\) sufficiently large (where \(\mathcal{X}_V\) is the characteristic function of the set \(V)\), then the following hold:
(a)
The map \(g\) is topologically ergodic.
(b)
The upper density \(\overline{\mu}(N_g(p,V))\) of \(N_g(p,V)\) is positive for any open subset \(V\neq\emptyset\) of \(S\), where \(N_g(p,V)=\{m\in\{0,1,\dots\}: g^m(p)\in V\}\).
(c)
There is a \(g\)-invariant Borel probability measure \(\nu\) having full support (i.e. \(\operatorname{supp}(\nu)=S)\).
(d)
Sensitivity of the map \(g\) implies positive lower density sensitivity, hence ergodical sensitivity.
(2)
If \(\underline{\mu}(N_g(A,B))=1\) for any two nonempty open subsets \(A,B\), then there exists \(\lambda>0\) satisfying \(\underline{\mu}( N_g(C,\lambda))=1\) for any nonempty open subset \(C\subset S\), where \(N_g(C,\lambda)=\{l\in\{0,1,\dots\}\): there exist \(a,b\in C\) with \(\tau(g^l(a), g^l(b))>\lambda\}\).

MSC:

37A25 Ergodicity, mixing, rates of mixing
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B02 Dynamics in general topological spaces
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[1] Abraham, C., Biau, G. & Cadre, B. [2002] “ Sensitive dependence on initial conditions,” Nonlinearity266, 420-431. · Zbl 1097.37034
[2] Akin, E. [1997] Recurrence in Topological Dynamics: Furstenberg Families and Ellis Actions (Springer, NY). · Zbl 0919.54033
[3] Auslander, J. & Yorke, J. [1980] “ Interval maps, factors of maps, and chaos,” Tohoku Math. J.32, 177-188. · Zbl 0448.54040
[4] Baladi, V. [2000] Positive Transfer Operators and Decay of Correlations, , Vol. 16 (World Scientific, Singapore). · Zbl 1012.37015
[5] Banks, J., Brooks, J., Cairns, G., Davis, G. & Stacey, P. [1992] “ On Devaney’s definition of chaos,” Amer. Math. Monthly99, 332-334. · Zbl 0758.58019
[6] Glasner, E. & Weiss, B. [1993] “ Sensitive dependence on initial conditions,” Nonlinearity6, 1067-1075. · Zbl 0790.58025
[7] Gu, R. B. [2007] “ The large deviations theorem and ergodicity,” Chaos Solit. Fract.34, 1387-1392. · Zbl 1152.37304
[8] He, L. F., Yan, X. H. & Wang, L. S. [2004] “ Weak-mixing implies sensitive dependence,” J. Math. Anal. Appl.299, 300-304. · Zbl 1057.28008
[9] Huang, Q. L., Shi, Y. M. & Zhang, L. J. [2015] “ Sensitivity of non-autonomous discrete dynamical systems,” Appl. Math. Lett.39, 31-34. · Zbl 1370.37031
[10] Kato, H. [1996] “ Everywhere chaotic homeomorphisms on manifolds and \(k\)-dimensional Menger manifolds,” Topol. Appl.72, 1-17. · Zbl 0859.54031
[11] Lardjane, S. [2006] “ On some stochastic properties in Devaney’s chaos,” Chaos Solit. Fract.28, 668-672. · Zbl 1106.37006
[12] Li, R. S. & Shi, Y. M. [2010] “ Several sufficient conditions for sensitive dependence on initial conditions,” Nonlin. Anal.72, 2716-2720. · Zbl 1180.37005
[13] Li, R. S. [2012a] “ A note on shadowing with chain transitivity,” Commun. Nonlin. Sci. Numer. Simulat.17, 2815-2823. · Zbl 1252.37007
[14] Li, R. S. [2012b] “ A note on stronger forms of sensitivity for dynamical systems,” Chaos Solit. Fract.45, 753-758. · Zbl 1263.37022
[15] Li, R. S. [2013] “ The large deviations theorem and ergodic sensitivity,” Commun. Nonlin. Sci. Numer. Simulat.18, 819-825. · Zbl 1258.37021
[16] Li, R. S. [2018] “ Several sufficient conditions for a map and a semi-flow to be ergodically sensitive,” Dyn. Syst.33, 358-360. · Zbl 1387.37005
[17] Li, R. S., Lu, T. X. & Waseem, A. [2019] “ Sensitivity and transitivity of systems satisfying the large deviations theorem in a sequence,” Int. J. Bifurcation and Chaos29, 1950125-1-9. · Zbl 1435.37016
[18] Li, R. S., Lu, T. X., Chen, G. R. & Liu, G. [2021] “ Some stronger forms of topological transitivity and sensitivity for a sequence of uniformly convergent continuous maps,” J. Math. Anal. Appl.494, 124443. · Zbl 1464.37015
[19] Liverani, C. [1995] “ Decay of correlations,” Ann. Math.142, 239-301. · Zbl 0871.58059
[20] Lu, T. X., Waseem, A. & Tang, X. [2019] “ Distributional chaoticity of C0-semigroup on a Frechet space,” Symmetry11, 345. · Zbl 1423.47003
[21] Martelli, M., Dang, M. & Seph, T. [1998] “Defining chaos,” Math. Mag.71, 112-122. · Zbl 1008.37014
[22] Moothathu, T. K. S. [2007] “ Stronger forms of sensitivity for dynamical systems,” Nonlinearity20, 2115-2126. · Zbl 1132.54023
[23] Niu, Y. X. [2009] “ The large deviations theorem and sensitivity,” Chaos Solit. Fract.42, 609-614. · Zbl 1198.37013
[24] Niu, Y. X., Su, S. B. & Zhou, B. D. [2015] “ Strong sensitivity of systems satisfying the large deviations theorem,” Int. J. Gen. Syst.44, 98-105. · Zbl 1332.54203
[25] Ruelle, D. & Takens, F. [1971] “ On the natural of turbulence,” Commun. Math. Phys.20, 167-192. · Zbl 0223.76041
[26] Vellekoop, M. & Berglund, R. [1994] “ On intervals, transitivity \(=\) chaos,” Amer. Math. Monthly101, 353-355. · Zbl 0886.58033
[27] Viana, M. [1997] Stochastic Dynamics of Deterministic Systems, (IMPA).
[28] Wang, H. Y. & Xiong, J. C. [2004] “ On the large deviations theorem and ergodicity,” Acta Math. Sin.47, 859-866 (in Chinese). · Zbl 1116.37006
[29] Wu, C., Xu, Z. J., Lin, W. & Ruan, J. [2005] “ Stochastic properties in Devaney’s chaos,” Chaos Solit. Fract.23, 1195-1199. · Zbl 1079.37026
[30] Wu, X. X. & Chen, G. R. [2016] “ On the large deviations theorem and ergodicity,” Commun. Nonlin. Sci. Numer. Simulat.30, 243-247. · Zbl 1489.37007
[31] Wu, X. X. & Wang, X. [2016] “ On the iteration properties of large deviations theorem,” Int. J. Bifurcation and Chaos26, 1650054-1-6. · Zbl 1336.37008
[32] Wu, X. X., Wang, X. & Chen, G. R. [2017] “ On the large deviations theorem of weaker types,” Int. J. Bifurcation and Chaos27, 1750127-1-12. · Zbl 1377.60044
[33] Wu, X. X., Ma, X., Zhu, Z. & Lu, T. X. [2018] “ Topological ergodic shadowing and chaos on uniform spaces,” Int. J. Bifurcation and Chaos28, 1850043-1-9. · Zbl 1387.37007
[34] Xu, Z. J., Lin, W. & Ruan, J. [2004] “ Decay of correlations implies chaos in the sense of Devaney,” Chaos Solit. Fract.22, 305-310. · Zbl 1060.37002
[35] Yong, L. S. [1986] “ Decay of correlations for piecewise expanding maps,” Ergod. Th. Dyn. Syst.6, 311-319. · Zbl 0633.58023
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