Kato’s chaos in set-valued discrete systems. (English) Zbl 1140.37305

Summary: We investigate the relationships between Kato’s chaoticity of a dynamical system \((X, f)\) and Kato’s chaoticity of the set-valued discrete system \((\mathcal K(X),\bar f)\) associated to \((X, f)\), where \(X\) is a compact metric space and \(f: X \rightarrow X\) is a continuous map. We show that Kato’s chaoticity of \((\mathcal K(X),\bar f)\) implies the Kato’s chaoticity of \((X, f)\) in general and \((X, f)\) is chaotic in the sense of Kato if and only if \((\mathcal K(X),\bar f)\) is Kato chaotic in \(w^e\)-topology. We also show that Ruelle-Takens’ chaoticity implies Kato’s chaoticity for a continuous map with a fixed point from a complete metric space without isolated point into itself.


37B99 Topological dynamics
Full Text: DOI


[1] Li, T. Y.; Yorke, J., Period three implies chaos, Amer Math Monthly, 82, 985-992 (1975) · Zbl 0351.92021
[2] Dvaney, R. L., An introduction to chaotic dynamical systems (1989), Addison-Wesley: Addison-Wesley Redwood City
[3] Banks, J.; Brooks, J.; Cairs, G.; Stacey, P., On the Devaney’s definition of chaos, Amer Math Monthly, 99, 332-334 (1992) · Zbl 0758.58019
[4] Ruelle, D.; Takens, F., On the natural of turbulence, Comm Math Phys, 20, 167-192 (1971) · Zbl 0223.76041
[5] Auslander, J.; Yorke, J., Interval maps, factors of maps, and chaos, Tohuko J Math, 32, 177-188 (1980) · Zbl 0448.54040
[6] Kato, H., Everywhere chaotic homeomorphisms on manifields and \(k\)-dimensional Menger manifolds, Topol Appl, 72, 1-17 (1996) · Zbl 0859.54031
[7] Roman-Flores, H., A note on transitivity in set-valued discrete systems, Chaos, Solitons & Fractals, 17, 99-104 (2003) · Zbl 1098.37008
[8] Nadler, S. B., Continuum theory, Pure Appl Math, 158 (1992)
[9] Xiong, J. C., Chaos on topological transitive systems, Sci China Ser A Math, 35, 302-311 (2005)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.