# 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 & Continuous Dynamical Systems - A, 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, 356 (2004), 21. Google Scholar [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. Google Scholar [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. doi: 10.1142/S0218127403007539. Google Scholar [4] J. Banks, Regular periodic decompositions for topologically transitive maps,, Ergodic Theory Dynam. Systems, 17 (1997), 505. doi: 10.1017/S0143385797069885. Google Scholar [5] J. Banks, Topological mapping properties defined by digraphs,, Discrete Contin. Dynam. Systems, 5 (1999), 83. doi: 10.3934/dcds.1999.5.83. Google Scholar [6] M. Barge and J. Martin, Dense orbits on the interval,, Michigan Math. J., 34 (1987), 3. doi: 10.1307/mmj/1029003477. Google Scholar [7] W. Bauer and K. Sigmund, Topological dynamics of transformations induced on the space of probability measures,, Monatsh. Math., 79 (1975), 81. doi: 10.1007/BF01585664. Google Scholar [8] F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs,, J. Reine Angew. Math., 547 (2002), 51. doi: 10.1515/crll.2002.053. Google Scholar [9] F. Blanchard and W. Huang, Entropy sets, weakly mixing sets and entropy capacity,, Discrete Contin. Dyn. Syst., 20 (2008), 275. Google Scholar [10] F. Blanchard, W. Huang and L. Snoha, Topological size of scrambled sets,, Colloq. Math., 110 (2008), 293. doi: 10.4064/cm110-2-3. Google Scholar [11] A. M. Blokh, On graph-realizable sets of periods,, J. Difference Equ. Appl., 9 (2003), 343. Google Scholar [12] R. Bowen, Topological entropy and axiom A,, in, 14 (1970). Google Scholar [13] J. Buzzi, Specification on the interval,, Trans. Amer. Math. Soc., 349 (1997), 2737. doi: 10.1090/S0002-9947-97-01873-4. Google Scholar [14] M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Spaces,", Lecture Notes in Mathematics, 527 (1976). Google Scholar [15] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation,, Math. Systems Theory, 1 (1967), 1. doi: 10.1007/BF01692494. Google Scholar [16] W. Huang and X. Ye, Devaney's chaos or 2-scattering implies Li-Yorke's chaos,, Topology Appl., 117 (2002), 259. doi: 10.1016/S0166-8641(01)00025-6. Google Scholar [17] A. Illanes and S. Nadler, "Hyperspaces,", Fundamentals and recent advances, 216 (1999). Google Scholar [18] A. Iwanik, Independence and scrambled sets for chaotic mappings,, The mathematical heritage of C. F. Gauss, (1991), 372. Google Scholar [19] H. Kato, On scrambled sets and a theorem of Kuratowski on independent sets,, Proc. Amer. Math. Soc., 126 (1998), 2151. doi: 10.1090/S0002-9939-98-04344-5. Google Scholar [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, 81 (1973), 65. Google Scholar [21] D. Kwietniak and M. Misiurewicz, Exact Devaney chaos and entropy,, Qual. Theory Dyn. Syst., 6 (2005), 169. doi: 10.1007/BF02972670. Google Scholar [22] S. H. Li, $\omega$-chaos and topological entropy,, Trans. Amer. Math. Soc., 339 (1993), 243. doi: 10.2307/2154217. Google Scholar [23] T. Y. Li and J. A. Yorke, Period three implies chaos,, Amer. Math. Monthly, 82 (1975), 985. doi: 10.2307/2318254. Google Scholar [24] G. Liao and L. Wang, Almost periodicity and distributional chaos,, in, (2000), 189. Google Scholar [25] E. Murinová, Generic chaos in metric spaces,, Acta Univ. M. Belii Ser. Math., 8 (2000), 43. Google Scholar [26] J.-H. Mai, Devaney's chaos implies existence of $s$-scrambled sets,, Proc. Amer. Math. Soc., 132 (2004), 2761. doi: 10.1090/S0002-9939-04-07514-8. Google Scholar [27] M. Málek, Distributional chaos for continuous mappings of the circle,, European Conference on Iteration Theory (Muszyna-Z\l ockie, 13 (1999), 205. Google Scholar [28] E. E. Moise, "Geometric Topology in Dimensions $2$ and $3$,", Graduate Texts in Mathematics, 47 (1977). Google Scholar [29] M. Morse and G. Hedlund, Symbolic dynamics II. Sturmian trajectories,, Amer. J. Math., 62 (1940), 1. doi: 10.2307/2371431. Google Scholar [30] J. Mycielski, Independent sets in topological algebras,, Fund. Math., 55 (1964), 139. Google Scholar [31] P. Oprocha, Specification properties and dense distributional chaos,, Discrete Contin. Dyn. Syst., 17 (2007), 821. doi: 10.3934/dcds.2007.17.821. Google Scholar [32] P. Oprocha, Distributional chaos revisited,, Trans. Amer. Math. Soc., 361 (2009), 4901. doi: 10.1090/S0002-9947-09-04810-7. Google Scholar [33] P. Oprocha and M. Štefánková, Specification property and distributional chaos almost everywhere,, Proc. Amer. Math. Soc., 136 (2008), 3931. doi: 10.1090/S0002-9939-08-09602-0. Google Scholar [34] P. Oprocha and G. Zhang, On local aspects of topological weak mixing in dimension one and beyond,, Studia Math., 202 (2011), 261. doi: 10.4064/sm202-3-4. Google Scholar [35] J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity,, Ann. of Math., 42 (1941), 874. doi: 10.2307/1968772. Google Scholar [36] J. Piórek, On the generic chaos in dynamical systems,, Univ. Iagel. Acta Math., 25 (1985), 293. Google Scholar [37] T. B. Rushing, "Topological Embeddings,", Pure and Applied Mathematics, 52 (1973). Google Scholar [38] S. Ruette, Dense chaos for continuous interval maps,, Nonlinearity, 18 (2005), 1691. doi: 10.1088/0951-7715/18/4/015. Google Scholar [39] S. Ruette, Chaos for continuous interval maps,, unpublished monograph., (). Google Scholar [40] S. Shao and X. Ye, $\mathcalF$-mixing and weak disjointness,, Topology Appl., 135 (2004), 231. doi: 10.1016/S0166-8641(03)00166-4. Google Scholar [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. doi: 10.2307/2154504. Google Scholar [42] J. Smítal, A chaotic function with some extremal properties,, Proc. Am. Math. Soc., 87 (1983), 54. Google Scholar [43] K. Sigmund, On dynamical systems with the specification property,, Trans. Amer. Math. Soc., 190 (1974), 285. doi: 10.1090/S0002-9947-1974-0352411-X. Google Scholar [44] A. Sklar and J. Smítal, Distributional chaos on compact metric spaces via specification properties,, J. Math. Anal. Appl., 241 (2000), 181. doi: 10.1006/jmaa.1999.6633. Google Scholar [45] J. Smítal and M. Štefánková, Omega-chaos almost everywhere,, Discrete Contin. Dyn. Syst., 9 (2003), 1323. doi: 10.3934/dcds.2003.9.1323. Google Scholar [46] L. Snoha, Generic chaos,, Comment. Math. Univ. Carolin., 31 (1990), 793. Google Scholar [47] L. Snoha, Dense chaos,, Comment. Math. Univ. Carolin., 33 (1992), 747. Google Scholar [48] J. C. Xiong and Z. G. Yang, Chaos caused by a topologically mixing map,, Dynamical systems and related topics (Nagoya, 9 (1991), 550. Google Scholar

show all references

##### References:
 [1] E. Akin, "Lectures on Cantor and Mycielski Sets for Dynamical Systems,", Chapel Hill Ergodic Theory Workshops, 356 (2004), 21. Google Scholar [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. Google Scholar [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. doi: 10.1142/S0218127403007539. Google Scholar [4] J. Banks, Regular periodic decompositions for topologically transitive maps,, Ergodic Theory Dynam. Systems, 17 (1997), 505. doi: 10.1017/S0143385797069885. Google Scholar [5] J. Banks, Topological mapping properties defined by digraphs,, Discrete Contin. Dynam. Systems, 5 (1999), 83. doi: 10.3934/dcds.1999.5.83. Google Scholar [6] M. Barge and J. Martin, Dense orbits on the interval,, Michigan Math. J., 34 (1987), 3. doi: 10.1307/mmj/1029003477. Google Scholar [7] W. Bauer and K. Sigmund, Topological dynamics of transformations induced on the space of probability measures,, Monatsh. Math., 79 (1975), 81. doi: 10.1007/BF01585664. Google Scholar [8] F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs,, J. Reine Angew. Math., 547 (2002), 51. doi: 10.1515/crll.2002.053. Google Scholar [9] F. Blanchard and W. Huang, Entropy sets, weakly mixing sets and entropy capacity,, Discrete Contin. Dyn. Syst., 20 (2008), 275. Google Scholar [10] F. Blanchard, W. Huang and L. Snoha, Topological size of scrambled sets,, Colloq. Math., 110 (2008), 293. doi: 10.4064/cm110-2-3. Google Scholar [11] A. M. Blokh, On graph-realizable sets of periods,, J. Difference Equ. Appl., 9 (2003), 343. Google Scholar [12] R. Bowen, Topological entropy and axiom A,, in, 14 (1970). Google Scholar [13] J. Buzzi, Specification on the interval,, Trans. Amer. Math. Soc., 349 (1997), 2737. doi: 10.1090/S0002-9947-97-01873-4. Google Scholar [14] M. Denker, C. Grillenberger and K. Sigmund, "Ergodic Theory on Compact Spaces,", Lecture Notes in Mathematics, 527 (1976). Google Scholar [15] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation,, Math. Systems Theory, 1 (1967), 1. doi: 10.1007/BF01692494. Google Scholar [16] W. Huang and X. Ye, Devaney's chaos or 2-scattering implies Li-Yorke's chaos,, Topology Appl., 117 (2002), 259. doi: 10.1016/S0166-8641(01)00025-6. Google Scholar [17] A. Illanes and S. Nadler, "Hyperspaces,", Fundamentals and recent advances, 216 (1999). Google Scholar [18] A. Iwanik, Independence and scrambled sets for chaotic mappings,, The mathematical heritage of C. F. Gauss, (1991), 372. Google Scholar [19] H. Kato, On scrambled sets and a theorem of Kuratowski on independent sets,, Proc. Amer. Math. Soc., 126 (1998), 2151. doi: 10.1090/S0002-9939-98-04344-5. Google Scholar [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, 81 (1973), 65. Google Scholar [21] D. Kwietniak and M. Misiurewicz, Exact Devaney chaos and entropy,, Qual. Theory Dyn. Syst., 6 (2005), 169. doi: 10.1007/BF02972670. Google Scholar [22] S. H. Li, $\omega$-chaos and topological entropy,, Trans. Amer. Math. Soc., 339 (1993), 243. doi: 10.2307/2154217. Google Scholar [23] T. Y. Li and J. A. Yorke, Period three implies chaos,, Amer. Math. Monthly, 82 (1975), 985. doi: 10.2307/2318254. Google Scholar [24] G. Liao and L. Wang, Almost periodicity and distributional chaos,, in, (2000), 189. Google Scholar [25] E. Murinová, Generic chaos in metric spaces,, Acta Univ. M. Belii Ser. Math., 8 (2000), 43. Google Scholar [26] J.-H. Mai, Devaney's chaos implies existence of $s$-scrambled sets,, Proc. Amer. Math. Soc., 132 (2004), 2761. doi: 10.1090/S0002-9939-04-07514-8. Google Scholar [27] M. Málek, Distributional chaos for continuous mappings of the circle,, European Conference on Iteration Theory (Muszyna-Z\l ockie, 13 (1999), 205. Google Scholar [28] E. E. Moise, "Geometric Topology in Dimensions $2$ and $3$,", Graduate Texts in Mathematics, 47 (1977). Google Scholar [29] M. Morse and G. Hedlund, Symbolic dynamics II. Sturmian trajectories,, Amer. J. Math., 62 (1940), 1. doi: 10.2307/2371431. Google Scholar [30] J. Mycielski, Independent sets in topological algebras,, Fund. Math., 55 (1964), 139. Google Scholar [31] P. Oprocha, Specification properties and dense distributional chaos,, Discrete Contin. Dyn. Syst., 17 (2007), 821. doi: 10.3934/dcds.2007.17.821. Google Scholar [32] P. Oprocha, Distributional chaos revisited,, Trans. Amer. Math. Soc., 361 (2009), 4901. doi: 10.1090/S0002-9947-09-04810-7. Google Scholar [33] P. Oprocha and M. Štefánková, Specification property and distributional chaos almost everywhere,, Proc. Amer. Math. Soc., 136 (2008), 3931. doi: 10.1090/S0002-9939-08-09602-0. Google Scholar [34] P. Oprocha and G. Zhang, On local aspects of topological weak mixing in dimension one and beyond,, Studia Math., 202 (2011), 261. doi: 10.4064/sm202-3-4. Google Scholar [35] J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity,, Ann. of Math., 42 (1941), 874. doi: 10.2307/1968772. Google Scholar [36] J. Piórek, On the generic chaos in dynamical systems,, Univ. Iagel. Acta Math., 25 (1985), 293. Google Scholar [37] T. B. Rushing, "Topological Embeddings,", Pure and Applied Mathematics, 52 (1973). Google Scholar [38] S. Ruette, Dense chaos for continuous interval maps,, Nonlinearity, 18 (2005), 1691. doi: 10.1088/0951-7715/18/4/015. Google Scholar [39] S. Ruette, Chaos for continuous interval maps,, unpublished monograph., (). Google Scholar [40] S. Shao and X. Ye, $\mathcalF$-mixing and weak disjointness,, Topology Appl., 135 (2004), 231. doi: 10.1016/S0166-8641(03)00166-4. Google Scholar [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. doi: 10.2307/2154504. Google Scholar [42] J. Smítal, A chaotic function with some extremal properties,, Proc. Am. Math. Soc., 87 (1983), 54. Google Scholar [43] K. Sigmund, On dynamical systems with the specification property,, Trans. Amer. Math. Soc., 190 (1974), 285. doi: 10.1090/S0002-9947-1974-0352411-X. Google Scholar [44] A. Sklar and J. Smítal, Distributional chaos on compact metric spaces via specification properties,, J. Math. Anal. Appl., 241 (2000), 181. doi: 10.1006/jmaa.1999.6633. Google Scholar [45] J. Smítal and M. Štefánková, Omega-chaos almost everywhere,, Discrete Contin. Dyn. Syst., 9 (2003), 1323. doi: 10.3934/dcds.2003.9.1323. Google Scholar [46] L. Snoha, Generic chaos,, Comment. Math. Univ. Carolin., 31 (1990), 793. Google Scholar [47] L. Snoha, Dense chaos,, Comment. Math. Univ. Carolin., 33 (1992), 747. Google Scholar [48] J. C. Xiong and Z. G. Yang, Chaos caused by a topologically mixing map,, Dynamical systems and related topics (Nagoya, 9 (1991), 550. Google Scholar
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