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On the integral systems with negative exponents
Recurrence properties and disjointness on the induced spaces
1. | Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, School of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, China, China, China |
References:
[1] |
J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies, 153, North-Holland, Amsterdam, 1988. |
[2] |
J. Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Th. and Dynam. Sys., 17 (1997), 505-529.
doi: 10.1017/S0143385797069885. |
[3] |
J. Banks, Chaos for induced hyperspace maps, Chaos Solitons Fractals, 25 (2005), 681-685.
doi: 10.1016/j.chaos.2004.11.089. |
[4] |
W. Bauer and K. Sigmund, Topological dynamics of transformations induced on the space of probability measures, Monatsh. Math., 79 (1975), 81-92.
doi: 10.1007/BF01585664. |
[5] |
A. Blokh and A. Fieldsteel, Sets that force recurrence, Proc. Amer. Math. Soc., 130 (2002), 3571-3578.
doi: 10.1090/S0002-9939-02-06349-9. |
[6] |
M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, 527, Springer-Verlag, Berlin-New York, 1976.
doi: 10.1007/BFb0082364. |
[7] |
P. Dong, S. Shao and X. Ye, Product recurrent properties, disjointness and weak disjointness, Israel J. of Math., 188 (2012), 463-507.
doi: 10.1007/s11856-011-0128-z. |
[8] |
H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory , 1 (1967), 1-49.
doi: 10.1007/BF01692494. |
[9] |
H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981. |
[10] |
E. Glasner and B. Weiss, Quasi-factors of zero-entropy systems, J. Amer. Math. Soc., 8 (1995), 665-686.
doi: 10.1090/S0894-0347-1995-1270579-5. |
[11] |
J. Guirao, K. Kwietniak, M. Lampart, P. Oprocha and A. Peris, Chaos on hyperspaces, Nonlinear Anal., 71 (2009), 1-8.
doi: 10.1016/j.na.2008.10.055. |
[12] |
W. Huang, H. Li and X. Ye, Family independence for topological and measurable dynamics, Trans. Amer. Math. Soc., 364 (2012), 5209-5242.
doi: 10.1090/S0002-9947-2012-05493-6. |
[13] |
W. Huang, K. Park and X. Ye, Topological disjointness from entropy zero systems, Bull. Soc. Math. France, 135 (2007), 259-282. |
[14] |
W. Huang and X. Ye, Dynamical systems disjoint from any minimal system, Trans. Amer. Math. Soc., 357 (2005), 669-694.
doi: 10.1090/S0002-9947-04-03540-8. |
[15] |
D. Kerr and H. Li, Dynamical entropy in Banach spaces, Invent. Math., 162 (2005), 649-686.
doi: 10.1007/s00222-005-0457-9. |
[16] |
D. Kerr and H. Li, Independence in topological and $C^*$-dynamics, Math. Ann., 338 (2007), 869-926.
doi: 10.1007/s00208-007-0097-z. |
[17] |
M. Komuro, The pseudo orbit tracing properties on the space of probability measures, Tokyo J. Math., 7 (1984), 461-468.
doi: 10.3836/tjm/1270151738. |
[18] |
E. Lehrer, Topological mixing and uniquely ergodic systems, Israel J. Math., 57 (1987), 239-255.
doi: 10.1007/BF02772176. |
[19] |
J. Li, Transitive points via Furstenberg family, Topology Appl., 158 (2011), 2221-2231.
doi: 10.1016/j.topol.2011.07.013. |
[20] |
J. Li, Equivalent conditions of Devaney chaos on the hyperspace, J. Univ. Sci. Technol. China, 44 (2014), 93-95. |
[21] |
Z. Lian, S. Shao and X. Ye, Weakly mixing, proximal topological models for ergodic systems and applications, preprint, arXiv:1407.1978v1. |
[22] |
Sam B. Nadler, Jr., Continuum Theory: An Introduction, Pure and Applied Mathematics, 158, Marcel Dekker Inc., New York, 1992. |
[23] |
P. Oprocha, Weak mixing and product recurrence, Ann. Inst. Fourier (Grenoble), 60 (2010), 1233-1257.
doi: 10.5802/aif.2553. |
[24] |
K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, 3, Academic Press, Inc. New York-London, 1967. |
[25] |
K. Petersen, Disjointness and weak mixing of minimal sets, Proc. Amer. Math. Soc., 24 (1970), 278-280.
doi: 10.1090/S0002-9939-1970-0250283-7. |
[26] |
H. Román-Flores, A note on transitivity in set-valued discrete systems, Chaos Solitons Fractals, 17 (2003), 99-104.
doi: 10.1016/S0960-0779(02)00406-X. |
[27] |
S. Shao, Dynamical Systems and Families, PhD thesis, University of Science and Technology of China, 2003. |
[28] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. |
show all references
References:
[1] |
J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies, 153, North-Holland, Amsterdam, 1988. |
[2] |
J. Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Th. and Dynam. Sys., 17 (1997), 505-529.
doi: 10.1017/S0143385797069885. |
[3] |
J. Banks, Chaos for induced hyperspace maps, Chaos Solitons Fractals, 25 (2005), 681-685.
doi: 10.1016/j.chaos.2004.11.089. |
[4] |
W. Bauer and K. Sigmund, Topological dynamics of transformations induced on the space of probability measures, Monatsh. Math., 79 (1975), 81-92.
doi: 10.1007/BF01585664. |
[5] |
A. Blokh and A. Fieldsteel, Sets that force recurrence, Proc. Amer. Math. Soc., 130 (2002), 3571-3578.
doi: 10.1090/S0002-9939-02-06349-9. |
[6] |
M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes in Mathematics, 527, Springer-Verlag, Berlin-New York, 1976.
doi: 10.1007/BFb0082364. |
[7] |
P. Dong, S. Shao and X. Ye, Product recurrent properties, disjointness and weak disjointness, Israel J. of Math., 188 (2012), 463-507.
doi: 10.1007/s11856-011-0128-z. |
[8] |
H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory , 1 (1967), 1-49.
doi: 10.1007/BF01692494. |
[9] |
H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981. |
[10] |
E. Glasner and B. Weiss, Quasi-factors of zero-entropy systems, J. Amer. Math. Soc., 8 (1995), 665-686.
doi: 10.1090/S0894-0347-1995-1270579-5. |
[11] |
J. Guirao, K. Kwietniak, M. Lampart, P. Oprocha and A. Peris, Chaos on hyperspaces, Nonlinear Anal., 71 (2009), 1-8.
doi: 10.1016/j.na.2008.10.055. |
[12] |
W. Huang, H. Li and X. Ye, Family independence for topological and measurable dynamics, Trans. Amer. Math. Soc., 364 (2012), 5209-5242.
doi: 10.1090/S0002-9947-2012-05493-6. |
[13] |
W. Huang, K. Park and X. Ye, Topological disjointness from entropy zero systems, Bull. Soc. Math. France, 135 (2007), 259-282. |
[14] |
W. Huang and X. Ye, Dynamical systems disjoint from any minimal system, Trans. Amer. Math. Soc., 357 (2005), 669-694.
doi: 10.1090/S0002-9947-04-03540-8. |
[15] |
D. Kerr and H. Li, Dynamical entropy in Banach spaces, Invent. Math., 162 (2005), 649-686.
doi: 10.1007/s00222-005-0457-9. |
[16] |
D. Kerr and H. Li, Independence in topological and $C^*$-dynamics, Math. Ann., 338 (2007), 869-926.
doi: 10.1007/s00208-007-0097-z. |
[17] |
M. Komuro, The pseudo orbit tracing properties on the space of probability measures, Tokyo J. Math., 7 (1984), 461-468.
doi: 10.3836/tjm/1270151738. |
[18] |
E. Lehrer, Topological mixing and uniquely ergodic systems, Israel J. Math., 57 (1987), 239-255.
doi: 10.1007/BF02772176. |
[19] |
J. Li, Transitive points via Furstenberg family, Topology Appl., 158 (2011), 2221-2231.
doi: 10.1016/j.topol.2011.07.013. |
[20] |
J. Li, Equivalent conditions of Devaney chaos on the hyperspace, J. Univ. Sci. Technol. China, 44 (2014), 93-95. |
[21] |
Z. Lian, S. Shao and X. Ye, Weakly mixing, proximal topological models for ergodic systems and applications, preprint, arXiv:1407.1978v1. |
[22] |
Sam B. Nadler, Jr., Continuum Theory: An Introduction, Pure and Applied Mathematics, 158, Marcel Dekker Inc., New York, 1992. |
[23] |
P. Oprocha, Weak mixing and product recurrence, Ann. Inst. Fourier (Grenoble), 60 (2010), 1233-1257.
doi: 10.5802/aif.2553. |
[24] |
K. R. Parthasarathy, Probability Measures on Metric Spaces, Probability and Mathematical Statistics, 3, Academic Press, Inc. New York-London, 1967. |
[25] |
K. Petersen, Disjointness and weak mixing of minimal sets, Proc. Amer. Math. Soc., 24 (1970), 278-280.
doi: 10.1090/S0002-9939-1970-0250283-7. |
[26] |
H. Román-Flores, A note on transitivity in set-valued discrete systems, Chaos Solitons Fractals, 17 (2003), 99-104.
doi: 10.1016/S0960-0779(02)00406-X. |
[27] |
S. Shao, Dynamical Systems and Families, PhD thesis, University of Science and Technology of China, 2003. |
[28] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982. |
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