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Furstenberg families and sensitivity. (English) Zbl 1187.37016
Summary: We introduce and study some concepts of sensitivity via Furstenberg families. A dynamical system \((X,f)\) is \({\mathcal F}\)-sensitive if there exists a positive \(\varepsilon\) such that for every \(x\in X\) and every open neighborhood \(U\) of \(x\) there exists \(y\in U\) such that the pair \((x,y)\) is not \({\mathcal F}\)-\(\varepsilon\)-asymptotic; that is, the time set \(\{n:d(f^n(x),f^n(y))>\varepsilon\}\) belongs to \({\mathcal F}\), where \({\mathcal F}\) is a Furstenberg family. A dynamical system \((X,f)\) is \(({\mathcal F}_1,{\mathcal F}_2)\)-sensitive if there is a positive \(\varepsilon\) such that every \(x\in X\) is a limit of points \(y\in X\) such that the pair \((x,y)\) is \({\mathcal F}_1\)-proximal but not \({\mathcal F}_2\)-\(\varepsilon\)-asymptotic; that is, the time set \(\{n:d(f^n(x),f^n(y))<\delta\}\) belongs to \({\mathcal F}_1\) for any positive \(\delta\) but the time set \(\{n:d(f^n(x),f^n(y))>\varepsilon\}\) belongs to \({\mathcal F}_2\), where \({\mathcal F}_1\) and \({\mathcal F}_2\) are Furstenberg families.

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B20 Notions of recurrence and recurrent behavior in dynamical systems
54H20 Topological dynamics (MSC2010)
58K15 Topological properties of mappings on manifolds
37B10 Symbolic dynamics
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[1] D. Ruelle, “Dynamical systems with turbulent behavior,” in Mathematical Problems in Theoretical Physics (Proc. Internat. Conf., Univ. Rome, Rome, 1977), vol. 80 of Lecture Notes in Physics, pp. 341-360, Springer, Berlin, Germany, 1978. · Zbl 0452.34038
[2] J. Guckenheimer, “Sensitive dependence to initial conditions for one-dimensional maps,” Communications in Mathematical Physics, vol. 70, no. 2, pp. 133-160, 1979. · Zbl 0429.58012 · doi:10.1007/BF01982351
[3] J. Auslander and J. A. Yorke, “Interval maps, factors of maps, and chaos,” The Tôhoku Mathematical Journal, vol. 32, no. 2, pp. 177-188, 1980. · Zbl 0448.54040 · doi:10.2748/tmj/1178229634
[4] R. Devaney, Chaotic Dynamical Systems, Addison-Wesley, Redwood City, Calif, USA, 1980. · Zbl 0463.58013
[5] C. Abraham, G. Biau, and B. Cadre, “On Lyapunov exponent and sensitivity,” Journal of Mathematical Analysis and Applications, vol. 290, no. 2, pp. 395-404, 2004. · Zbl 1034.37019 · doi:10.1016/j.jmaa.2003.10.029
[6] E. Akin and S. Kolyada, “Li-Yorke sensitivity,” Nonlinearity, vol. 16, no. 4, pp. 1421-1433, 2003. · Zbl 1045.37004 · doi:10.1088/0951-7715/16/4/313
[7] F. Blanchard, E. Glasner, S. Kolyada, and A. Maass, “On Li-Yorke pairs,” Journal für die Reine und Angewandte Mathematik, vol. 547, pp. 51-68, 2002. · Zbl 1059.37006 · doi:10.1515/crll.2002.053
[8] T. Y. Li and J. A. Yorke, “Period three implies chaos,” The American Mathematical Monthly, vol. 82, no. 10, pp. 985-992, 1975. · Zbl 0351.92021 · doi:10.2307/2318254
[9] B. Schweizer and J. Smítal, “Measures of chaos and a spectral decomposition of dynamical systems on the interval,” Transactions of the American Mathematical Society, vol. 344, no. 2, pp. 737-754, 1994. · Zbl 0812.58062 · doi:10.2307/2154504
[10] E. Akin, Recurrence in Topological Dynamics, Furstenberg Families and Ellis Actions, The University Series in Mathematics, Plenum Press, New York, NY, USA, 1997. · Zbl 0919.54033
[11] W. Huang, S. Shao, and X. Ye, “Mixing and proximal cells along sequences,” Nonlinearity, vol. 17, no. 4, pp. 1245-1260, 2004. · Zbl 1055.37014 · doi:10.1088/0951-7715/17/4/006
[12] S. Shao, “Proximity and distality via Furstenberg families,” Topology and Its Applications, vol. 153, no. 12, pp. 2055-2072, 2006. · Zbl 1099.37006 · doi:10.1016/j.topol.2005.07.012
[13] J. C. Xiong, J. Lü, and F. Tan, “Furstenberg familes and chaos,” Science in China Series A, vol. 50, pp. 1352-1333, 2007. · Zbl 1136.54025 · doi:10.1007/s11425-007-0052-1
[14] F. Tan and J. Xiong, “Chaos via Furstenberg family couple,” Topology and Its Applications, vol. 156, no. 3, pp. 525-532, 2009. · Zbl 1161.37019 · doi:10.1016/j.topol.2008.08.006
[15] W. Huang and X. Ye, “Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos,” Topology and Its Applications, vol. 117, no. 3, pp. 259-272, 2002. · Zbl 0997.54061 · doi:10.1016/S0166-8641(01)00025-6
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