American Institute of Mathematical Sciences

September  2011, 31(3): 797-825. doi: 10.3934/dcds.2011.31.797

Coherent lists and chaotic sets

 1 Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków

Received  April 2010 Revised  July 2011 Published  August 2011

In this article we apply (recently extended by Kato and Akin) an elegant method of Iwanik (which adopts independence relations of Kuratowski and Mycielski) in the construction of various chaotic sets. We provide ''easy to track'' proofs of some known facts and establish new results as well. The main advantage of the presented approach is that it is easy to verify each step of the proof, when previously it was almost impossible to go into all the details of the construction (usually performed as an inductive procedure). Furthermore, we are able extend known results on chaotic sets in an elegant way. Scrambled, distributionally scrambled and chaotic sets with relation to various notions of mixing are considered.
Citation: Piotr Oprocha. Coherent lists and chaotic sets. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 797-825. doi: 10.3934/dcds.2011.31.797
References:
 [1] E. Akin, "Lectures on Cantor and Mycielski Sets for Dynamical Systems," Chapel Hill Ergodic Theory Workshops, Contemp. Math., 356, Amer. Math. Soc., Providence, RI, 2004, 21-79. [2] Ll. Alsedà, M. A. del Río and J. A. Rodríguez, Transitivity and dense periodicity for graph maps, J. Difference Equ. Appl., 9 (2003), 577-598. [3] F. Balibrea, B. Schweizer, A. Sklar and J. Smítal, Generalized specification property and distributional chaos, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1683-1694. doi: 10.1142/S0218127403007539. [4] J. Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Theory Dynam. Systems, 17 (1997), 505-529. doi: 10.1017/S0143385797069885. [5] J. Banks, Topological mapping properties defined by digraphs, Discrete Contin. Dynam. Systems, 5 (1999), 83-92. doi: 10.3934/dcds.1999.5.83. [6] M. Barge and J. Martin, Dense orbits on the interval, Michigan Math. J., 34 (1987), 3-11. doi: 10.1307/mmj/1029003477. [7] 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. [8] F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68. doi: 10.1515/crll.2002.053. [9] F. Blanchard and W. Huang, Entropy sets, weakly mixing sets and entropy capacity, Discrete Contin. Dyn. Syst., 20 (2008), 275-311. [10] F. Blanchard, W. Huang and L. Snoha, Topological size of scrambled sets, Colloq. Math., 110 (2008), 293-361. doi: 10.4064/cm110-2-3. [11] A. M. Blokh, On graph-realizable sets of periods, J. Difference Equ. Appl., 9 (2003), 343-357. [12] R. Bowen, Topological entropy and axiom A, in "Global Analysis," Proceedings of Symposia on Pure Mathematics, 14, Am. Math. Soc., Providence, RI, 1970. [13] J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754. doi: 10.1090/S0002-9947-97-01873-4. [14] M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Spaces," Lecture Notes in Mathematics, 527, Springer-Verlag, Berlin-New York, 1976. [15] 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. [16] W. Huang and X. Ye, Devaney's chaos or 2-scattering implies Li-Yorke's chaos, Topology Appl., 117 (2002), 259-272. doi: 10.1016/S0166-8641(01)00025-6. [17] A. Illanes and S. Nadler, "Hyperspaces," Fundamentals and recent advances, Monographs and Textbooks in Pure and Applied Mathematics, 216, Marcel Dekker, Inc., New York, 1999. [18] A. Iwanik, Independence and scrambled sets for chaotic mappings, The mathematical heritage of C. F. Gauss, 372-378, World Sci. Publ., River Edge, NJ, 1991. [19] H. Kato, On scrambled sets and a theorem of Kuratowski on independent sets, Proc. Amer. Math. Soc., 126 (1998), 2151-2157. doi: 10.1090/S0002-9939-98-04344-5. [20] K. Kuratowski, Applications of the Baire-category method to the problem of independent sets, Collection of articles dedicated to Andrzej Mostowski on the occasion of his sixtieth birthday, Fund. Math., 81 (1973), 65-72. [21] D. Kwietniak and M. Misiurewicz, Exact Devaney chaos and entropy, Qual. Theory Dyn. Syst., 6 (2005), 169-179. doi: 10.1007/BF02972670. [22] S. H. Li, $\omega$-chaos and topological entropy, Trans. Amer. Math. Soc., 339 (1993), 243-249. doi: 10.2307/2154217. [23] T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. doi: 10.2307/2318254. [24] G. Liao and L. Wang, Almost periodicity and distributional chaos, in "Foundations of Computational Mathematics" (Hong Kong, 2000), 189-210, World Sci. Publ., River Edge, NJ, 2002. [25] E. Murinová, Generic chaos in metric spaces, Acta Univ. M. Belii Ser. Math., 8 (2000), 43-50. [26] J.-H. Mai, Devaney's chaos implies existence of $s$-scrambled sets, Proc. Amer. Math. Soc., 132 (2004), 2761-2767. doi: 10.1090/S0002-9939-04-07514-8. [27] M. Málek, Distributional chaos for continuous mappings of the circle, European Conference on Iteration Theory (Muszyna-Z\l ockie, 1998), Ann. Math. Sil., 13 (1999), 205-210. [28] E. E. Moise, "Geometric Topology in Dimensions $2$ and $3$," Graduate Texts in Mathematics, 47, Springer-Verlag, New York-Heidelberg, 1977. [29] M. Morse and G. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42. doi: 10.2307/2371431. [30] J. Mycielski, Independent sets in topological algebras, Fund. Math., 55 (1964), 139-147. [31] P. Oprocha, Specification properties and dense distributional chaos, Discrete Contin. Dyn. Syst., 17 (2007), 821-833. doi: 10.3934/dcds.2007.17.821. [32] P. Oprocha, Distributional chaos revisited, Trans. Amer. Math. Soc., 361 (2009), 4901-4925. doi: 10.1090/S0002-9947-09-04810-7. [33] P. Oprocha and M. Štefánková, Specification property and distributional chaos almost everywhere, Proc. Amer. Math. Soc., 136 (2008), 3931-3940. doi: 10.1090/S0002-9939-08-09602-0. [34] P. Oprocha and G. Zhang, On local aspects of topological weak mixing in dimension one and beyond, Studia Math., 202 (2011), 261-288. doi: 10.4064/sm202-3-4. [35] J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math., 42 (1941), 874-920. doi: 10.2307/1968772. [36] J. Piórek, On the generic chaos in dynamical systems, Univ. Iagel. Acta Math., 25 (1985), 293-298. [37] T. B. Rushing, "Topological Embeddings," Pure and Applied Mathematics, 52, Academic Press, New York-London, 1973. [38] S. Ruette, Dense chaos for continuous interval maps, Nonlinearity, 18 (2005), 1691-1698. doi: 10.1088/0951-7715/18/4/015. [39] S. Ruette, Chaos for continuous interval maps, unpublished monograph. [40] S. Shao and X. Ye, $\mathcal{F}$-mixing and weak disjointness, Topology Appl., 135 (2004), 231-247. doi: 10.1016/S0166-8641(03)00166-4. [41] B. Schweizer and J. Smítal, Measure of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., 344 (1994), 737-754. doi: 10.2307/2154504. [42] J. Smítal, A chaotic function with some extremal properties, Proc. Am. Math. Soc., 87 (1983), 54-56. [43] K. Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc., 190 (1974), 285-299. doi: 10.1090/S0002-9947-1974-0352411-X. [44] A. Sklar and J. Smítal, Distributional chaos on compact metric spaces via specification properties, J. Math. Anal. Appl., 241 (2000), 181-188. doi: 10.1006/jmaa.1999.6633. [45] J. Smítal and M. Štefánková, Omega-chaos almost everywhere, Discrete Contin. Dyn. Syst., 9 (2003), 1323-1327. doi: 10.3934/dcds.2003.9.1323. [46] L. Snoha, Generic chaos, Comment. Math. Univ. Carolin., 31 (1990), 793-810. [47] L. Snoha, Dense chaos, Comment. Math. Univ. Carolin., 33 (1992), 747-752. [48] J. C. Xiong and Z. G. Yang, Chaos caused by a topologically mixing map, Dynamical systems and related topics (Nagoya, 1990), Adv. Ser. Dynam. Systems, 9, World Sci. Publ., River Edge, NJ, (1991), 550-572.

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References:
 [1] E. Akin, "Lectures on Cantor and Mycielski Sets for Dynamical Systems," Chapel Hill Ergodic Theory Workshops, Contemp. Math., 356, Amer. Math. Soc., Providence, RI, 2004, 21-79. [2] Ll. Alsedà, M. A. del Río and J. A. Rodríguez, Transitivity and dense periodicity for graph maps, J. Difference Equ. Appl., 9 (2003), 577-598. [3] F. Balibrea, B. Schweizer, A. Sklar and J. Smítal, Generalized specification property and distributional chaos, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 1683-1694. doi: 10.1142/S0218127403007539. [4] J. Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Theory Dynam. Systems, 17 (1997), 505-529. doi: 10.1017/S0143385797069885. [5] J. Banks, Topological mapping properties defined by digraphs, Discrete Contin. Dynam. Systems, 5 (1999), 83-92. doi: 10.3934/dcds.1999.5.83. [6] M. Barge and J. Martin, Dense orbits on the interval, Michigan Math. J., 34 (1987), 3-11. doi: 10.1307/mmj/1029003477. [7] 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. [8] F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68. doi: 10.1515/crll.2002.053. [9] F. Blanchard and W. Huang, Entropy sets, weakly mixing sets and entropy capacity, Discrete Contin. Dyn. Syst., 20 (2008), 275-311. [10] F. Blanchard, W. Huang and L. Snoha, Topological size of scrambled sets, Colloq. Math., 110 (2008), 293-361. doi: 10.4064/cm110-2-3. [11] A. M. Blokh, On graph-realizable sets of periods, J. Difference Equ. Appl., 9 (2003), 343-357. [12] R. Bowen, Topological entropy and axiom A, in "Global Analysis," Proceedings of Symposia on Pure Mathematics, 14, Am. Math. Soc., Providence, RI, 1970. [13] J. Buzzi, Specification on the interval, Trans. Amer. Math. Soc., 349 (1997), 2737-2754. doi: 10.1090/S0002-9947-97-01873-4. [14] M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Spaces," Lecture Notes in Mathematics, 527, Springer-Verlag, Berlin-New York, 1976. [15] 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. [16] W. Huang and X. Ye, Devaney's chaos or 2-scattering implies Li-Yorke's chaos, Topology Appl., 117 (2002), 259-272. doi: 10.1016/S0166-8641(01)00025-6. [17] A. Illanes and S. Nadler, "Hyperspaces," Fundamentals and recent advances, Monographs and Textbooks in Pure and Applied Mathematics, 216, Marcel Dekker, Inc., New York, 1999. [18] A. Iwanik, Independence and scrambled sets for chaotic mappings, The mathematical heritage of C. F. Gauss, 372-378, World Sci. Publ., River Edge, NJ, 1991. [19] H. Kato, On scrambled sets and a theorem of Kuratowski on independent sets, Proc. Amer. Math. Soc., 126 (1998), 2151-2157. doi: 10.1090/S0002-9939-98-04344-5. [20] K. Kuratowski, Applications of the Baire-category method to the problem of independent sets, Collection of articles dedicated to Andrzej Mostowski on the occasion of his sixtieth birthday, Fund. Math., 81 (1973), 65-72. [21] D. Kwietniak and M. Misiurewicz, Exact Devaney chaos and entropy, Qual. Theory Dyn. Syst., 6 (2005), 169-179. doi: 10.1007/BF02972670. [22] S. H. Li, $\omega$-chaos and topological entropy, Trans. Amer. Math. Soc., 339 (1993), 243-249. doi: 10.2307/2154217. [23] T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. doi: 10.2307/2318254. [24] G. Liao and L. Wang, Almost periodicity and distributional chaos, in "Foundations of Computational Mathematics" (Hong Kong, 2000), 189-210, World Sci. Publ., River Edge, NJ, 2002. [25] E. Murinová, Generic chaos in metric spaces, Acta Univ. M. Belii Ser. Math., 8 (2000), 43-50. [26] J.-H. Mai, Devaney's chaos implies existence of $s$-scrambled sets, Proc. Amer. Math. Soc., 132 (2004), 2761-2767. doi: 10.1090/S0002-9939-04-07514-8. [27] M. Málek, Distributional chaos for continuous mappings of the circle, European Conference on Iteration Theory (Muszyna-Z\l ockie, 1998), Ann. Math. Sil., 13 (1999), 205-210. [28] E. E. Moise, "Geometric Topology in Dimensions $2$ and $3$," Graduate Texts in Mathematics, 47, Springer-Verlag, New York-Heidelberg, 1977. [29] M. Morse and G. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42. doi: 10.2307/2371431. [30] J. Mycielski, Independent sets in topological algebras, Fund. Math., 55 (1964), 139-147. [31] P. Oprocha, Specification properties and dense distributional chaos, Discrete Contin. Dyn. Syst., 17 (2007), 821-833. doi: 10.3934/dcds.2007.17.821. [32] P. Oprocha, Distributional chaos revisited, Trans. Amer. Math. Soc., 361 (2009), 4901-4925. doi: 10.1090/S0002-9947-09-04810-7. [33] P. Oprocha and M. Štefánková, Specification property and distributional chaos almost everywhere, Proc. Amer. Math. Soc., 136 (2008), 3931-3940. doi: 10.1090/S0002-9939-08-09602-0. [34] P. Oprocha and G. Zhang, On local aspects of topological weak mixing in dimension one and beyond, Studia Math., 202 (2011), 261-288. doi: 10.4064/sm202-3-4. [35] J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math., 42 (1941), 874-920. doi: 10.2307/1968772. [36] J. Piórek, On the generic chaos in dynamical systems, Univ. Iagel. Acta Math., 25 (1985), 293-298. [37] T. B. Rushing, "Topological Embeddings," Pure and Applied Mathematics, 52, Academic Press, New York-London, 1973. [38] S. Ruette, Dense chaos for continuous interval maps, Nonlinearity, 18 (2005), 1691-1698. doi: 10.1088/0951-7715/18/4/015. [39] S. Ruette, Chaos for continuous interval maps, unpublished monograph. [40] S. Shao and X. Ye, $\mathcal{F}$-mixing and weak disjointness, Topology Appl., 135 (2004), 231-247. doi: 10.1016/S0166-8641(03)00166-4. [41] B. Schweizer and J. Smítal, Measure of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc., 344 (1994), 737-754. doi: 10.2307/2154504. [42] J. Smítal, A chaotic function with some extremal properties, Proc. Am. Math. Soc., 87 (1983), 54-56. [43] K. Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc., 190 (1974), 285-299. doi: 10.1090/S0002-9947-1974-0352411-X. [44] A. Sklar and J. Smítal, Distributional chaos on compact metric spaces via specification properties, J. Math. Anal. Appl., 241 (2000), 181-188. doi: 10.1006/jmaa.1999.6633. [45] J. Smítal and M. Štefánková, Omega-chaos almost everywhere, Discrete Contin. Dyn. Syst., 9 (2003), 1323-1327. doi: 10.3934/dcds.2003.9.1323. [46] L. Snoha, Generic chaos, Comment. Math. Univ. Carolin., 31 (1990), 793-810. [47] L. Snoha, Dense chaos, Comment. Math. Univ. Carolin., 33 (1992), 747-752. [48] J. C. Xiong and Z. G. Yang, Chaos caused by a topologically mixing map, Dynamical systems and related topics (Nagoya, 1990), Adv. Ser. Dynam. Systems, 9, World Sci. Publ., River Edge, NJ, (1991), 550-572.
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