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Several sufficient conditions for a map and a semi-flow to be ergodically sensitive. (English) Zbl 1387.37005
Summary: In this paper, we are concerned solely with the ergodic sensitivity for maps (resp. semi-flows), where these maps and semi-flows may not be continuous (in topology). A few new sufficient conditions under which a map (resp. a semi-flow) on a metric space is ergodically sensitive are presented, where such maps and semi-flows may not be continuous (in topology). In particular, we prove that the topologically strong ergodicity of a measure-preserving map (resp. a measure-preserving semi-flow) on a metric probability space with a fully supported measure implies its ergodical sensitivity.

MSC:
 37A05 Dynamical aspects of measure-preserving transformations 37A25 Ergodicity, mixing, rates of mixing 54H20 Topological dynamics (MSC2010) 28D05 Measure-preserving transformations
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 [1] Abraham, C.; Biau, G.; Cadre, B., chaotic properties of mapping on a probability space, J. Math. Anal. Appl., 266, 420-431, (2002) · Zbl 1097.37034 [2] Dastjerdi, D. Ahmadi; Hosseini, M., sub-shadowings, Nonlinear Anal., 72, 3759-3766, (2010) · Zbl 1197.54046 [3] Bank, J.; Brooks, J.; Cairns, G.; Davis, G.; Stacey, D., on Devaney’s definition of chaos, Amer. Math. Monthly, 99, 332-334, (1992) · Zbl 0758.58019 [4] Glasner, E.; Weiss, B., sensitive dependence on initial conditions, Nonlinearity, 6, 1067-1075, (1993) · Zbl 0790.58025 [5] Gu, R., the large deviations theorem and ergodicity, Chaos Solitons Fractals, 34, 1387-1392, (2007) · Zbl 1152.37304 [6] Hasselblatt, B.; Katok, A., A First Course in Dynamics, (2003), Cambridge University Press, New York · Zbl 1027.37001 [7] He, L.; Yan, X.; Wang, L., weak-mixing implies sensitive dependence, J. Math. Anal. Appl., 299, 300-304, (2004) · Zbl 1057.28008 [8] Kato, H., everywhere chaotic homeomorphisms on manifolds and k −dimensional merger manifolds, Topology Appl., 72, 1-17, (1996) · Zbl 0859.54031 [9] Lardjane, S., on some stochastic properties in Devaney’s chaos, Chaos Solitons Fractals, 28, 668-672, (2006) · Zbl 1106.37006 [10] Li, R., A note on stronger forms of sensitivity for dynamical systems, Chaos Solitons Fractals, 45, 753-758, (2012) · Zbl 1263.37022 [11] Li, R., the large deviations theorem and ergodic sensitivity, Commun. Nonlinear Sci. Numer. Simul., 18, 819-825, (2013) · Zbl 1258.37021 [12] Li, R., A note on shadowing with chain transitivity, Commun. Nonlinear Sci. Numer. Simul., 17, 2815-2823, (2012) · Zbl 1252.37007 [13] Li, R.; Shi, Y., several sufficient conditions for sensitive dependence on initial conditions, Nonlinear Anal., 72, 2716-2720, (2010) · Zbl 1180.37005 [14] Nemiskii, V.V.; Stepanov, V.V., Qualitative Theory of Ordinary Differential Equations, (1960), Princeton Univ. Press, Princeton, NJ [15] Niu, Y., the average-shadowing property and strong ergodicity, J. Math. Anal. Appl., 376, 528-534, (2011) · Zbl 1210.37016 [16] Niu, Y., the large deviations theorem and sensitivity, Chaos Solitons Fractals, 42, 609-614, (2009) · Zbl 1198.37013 [17] Niu, Y.; Su, S., on strong ergodicity and chaoticity of systems with the asymptotic average shadowing property, Chaos Solitons Fractals, 44, 429-432, (2011) · Zbl 1225.37031 [18] Niu, Y.; Su, S.; Zhou, and B., strong sensitivity of systems satisfying the large deviations theorem, Int. J. Gen. Syst., 44, 98-105, (2015) · Zbl 1332.54203 [19] Niu, Y.; Wang, Y.; Su, S., the asymptotic average shadowing property and strong ergodicity, Chaos Solitons Fractals, 53, 34-38, (2013) · Zbl 1339.37003 [20] Peterson, K., Ergodic Theory, (1983), Cambridge University Press, New York [21] Walters, P., An Introduction to Ergodic Theory, (1982), Springer-Verlag, New York · Zbl 0475.28009 [22] Wang, Y.; Niu, Y., strong ergodicity of systems with the average shadowing property, Dynam. Sys.-An Int. J., 29, 18-23, (2014) · Zbl 1291.37007 [23] Wang, H.; Xiong, J., some properties of topologically ergodic maps, Acta. Math. Sin., 47, 859-866, (2004) · Zbl 1116.37006 [24] Wu, X.; Chen, G., on the large deviations theorem and ergodicity, Commun. Nonlin. Sci. Numer. Simulat., 30, 243-247, (2016) [25] Xiong, J.; Yang, Z., Chaos caused by a topologically mixing maps, in Dynamical Systems and Related Topics, (1992), World Scientific Press, Singapore [26] Yang, R., topologically ergodic maps, Acta. Math. Sinica, 44, 1063-1068, (2001) · Zbl 1012.37007 [27] Yang, R., topological ergodicity and topological double ergodicity, Acta. Math. Sinica, 46, 555-560, (2003) · Zbl 1047.54023
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