Browse > Article
http://dx.doi.org/10.4134/JKMS.2014.51.4.837

TRANSITIVITY, TWO-SIDED LIMIT SHADOWING PROPERTY AND DENSE ω-CHAOS  

Oprocha, Piotr (AGH University of Science and Technology Faculty of Applied Mathematics, Institute of Mathematics Polish Academy of Sciences)
Publication Information
Journal of the Korean Mathematical Society / v.51, no.4, 2014 , pp. 837-851 More about this Journal
Abstract
We consider ${\omega}$-chaos as defined by S. H. Li in 1993. We show that c-dense ${\omega}$-scrambled sets are present in every transitive system with two-sided limit shadowing property (TSLmSP) and that every transitive map on topological graph has a dense Mycielski ${\omega}$-scrambled set. As a preliminary step, we provide a characterization of dynamical properties of maps with TSLmSP.
Keywords
shadowing property; limit shadowing; pseudo-orbit; asymptotic tracing; minimal set; ${\omega}$-chaos; topological graph;
Citations & Related Records
연도 인용수 순위
  • Reference
1 S. M. Srivastava, A Course on Borel Sets, Graduate Texts in Mathematics, vol. 180, Springer-Verlag, New York, 1998.
2 S. Yu. Pilyugin, Shadowing in Dynamical Systems, Springer-Verlag, Berlin, 1999.
3 K. Sakai, Diffeomorphisms with the s-limit shadowing property, Dyn. Syst. 27 (2012), no. 4, 403-410.   DOI
4 J. Smital and M. Stefankova, Omega-chaos almost everywhere, Discrete Contin. Dyn. Syst. 9 (2003), no. 5, 1323-1327.   DOI
5 D. Kwietniak and P. Oprocha, A note on the average shadowing property for expansive maps, Topology Appl. 159 (2012), no. 1, 19-27.   DOI   ScienceOn
6 G. Haranczyk, D. Kwietniak, and P. Oprocha, A note on transitivity, sensitivity and chaos for graph maps, J. Difference Equ. Appl. 17 (2011), no. 10, 1549-1553.   DOI   ScienceOn
7 G. Haranczyk, D. Kwietniak, and P. Oprocha, Topological structure and entropy of mixing graph maps, Ergodic Theory Dynam. Systems (2013); doi: 10.1017/etds.2013.6   DOI   ScienceOn
8 M. Kulczycki, D. Kwietniak, and P. Oprocha, On almost specification and average shad-owing properties, preprint, 2013.
9 M. Lampart and P. Oprocha, Shift spaces, ${\omega}$-chaos and specification property, Topology Appl. 156 (2009), no. 18, 2979-2985.   DOI   ScienceOn
10 K. Lee and K. Sakai, Various shadowing properties and their equivalence, Discrete Contin. Dyn. Syst. 13 (2005), no. 2, 533-540.   DOI
11 S. H. Li, ${\omega}$-chaos and topological entropy, Trans. Amer. Math. Soc. 339 (1993), no. 1, 243-249.
12 J. Llibre and M. Misiurewicz, Horseshoes, entropy and periods for graph maps, Topology 32 (1993), no. 3, 649-664.   DOI   ScienceOn
13 G. Lu, K. Lee, and M. Lee, Generic diffeomorphisms with weak limit shadowing, Adv. Difference Equ. 2013 (2013), article id: 27, doi: 10.1186/1687-1847-2013-27.   DOI   ScienceOn
14 J. H. Mai and X. Ye, The structure of pointwise recurrent maps having the pseudo orbit tracing property, Nagoya Math. J. 166 (2002), 83-92.   DOI
15 J. Mycielski, Independent sets in topological algebras, Fund. Math. 55 (1964), 137-147.
16 T. K. S. Moothathu, Implications of pseudo-orbit tracing property for continuous maps on compacta, Top. Appl. 158 (2011), no. 16, 2232-2239.   DOI   ScienceOn
17 T. K. S. Moothathu, Syndetically proximal pairs, J. Math. Anal. Appl. 379 (2011), no. 2, 656-663.   DOI   ScienceOn
18 T. K. S. Moothathu and P. Oprocha, Shadowing entropy and minimal subsystems, Monatsh. Math. 172 (2013), no. 3-4, 357-378.   DOI
19 R. Pikula, On some notions of chaos in dimension zero, Colloq. Math. 107 (2007), no. 2, 167-177.   DOI
20 K. Palmer, Shadowing in Dynamical Systems, Kluwer Academic Publishers, Dordrecht, 2000.
21 J. Banks and B. Trotta, Weak mixing implies mixing for maps on topological graphs, J. Difference Equ. Appl. 11 (2005), no. 12, 1071-1080.   DOI   ScienceOn
22 Ll. Alseda, M. A. del Rio, and J. A. Rodriguez, Transitivity and dense periodicity for graph maps, J. Difference Equ. Appl. 9 (2003), no. 6, 577-598.   DOI   ScienceOn
23 J. Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Theory Dynam. Systems 17 (1997), no. 3, 505-529.   DOI
24 J. Bobok, On multidimensional ${\omega}$-chaos, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 16 (2006), no. 3, 737-740.   DOI   ScienceOn
25 B. Carvalho, Hyperbolicity, transitivity and the two-sided limit shadowing property, Proc. AMS, to appear.
26 B. Carvalho and D. Kwietniak, On homeomorphisms with the two-sided limit shadowing property, preprint, arXiv:1402.0674.
27 M. Dirbak, L. Snoha, and V. Spitalsky, Minimality, transitivity, mixing and topological entropy on spaces with a free interval, Ergodic Theory Dynam. Systems 33 (2013), no. 6, 1786-1812.   DOI
28 S. Yu. Pilyugin, Sets of dynamical systems with various limit shadowing properties, J. Dynam. Differential Equations 19 (2007), no. 3, 747-775.   DOI
29 Ll. Alseda, M. A. del Rio, and J. A. Rodriguez, A splitting theorem for transitive maps, J. Math. Anal. Appl. 232 (1999), no. 2, 359-375.   DOI   ScienceOn