×

Chaotic dynamics of continuous-time topological semi-flows on Polish spaces. (English) Zbl 1320.34069

In the paper under review a semiflow \((\mathbb{R}_+,X,f)\), shortly \((\mathbb{R}_+,X)\), is a continuous monoid action \(f:\mathbb{R}_+\times X\to X\), \((t,x)\mapsto f(t,x)=:tx\) for every \(t\in \mathbb{R}_+, x\in X\), of the additive monoid \(\mathbb{R}_+=[0,\infty)\) on a Polish space \((X,d)\). The main result of the paper is the following theorem: Let \((\mathbb{R}_+,X)\) be a semiflow which is non-minimal, point-transitive and such that the set of points \(x\in X\) whose orbit-closure is a compact minimal subset of \(X\) is dense in \(X\). Then the semiflow \((\mathbb{R}_+,X)\) is sensitive in the following sense: there is a number \(c>0\) such that for every \(x\in X\) there exists a dense \(G_\delta\)-set \(S_c(x)\subset X\) with the property \(\limsup_{t\to\infty} d(tx,ty)\geq c\) for each \(y\in S_c(x)\).
The paper is well written and interesting.

MSC:

34C28 Complex behavior and chaotic systems of ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
54H20 Topological dynamics (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Banks, J.; Brooks, J.; Cairns, G.; Davis, G.; Stacey, P., On Devaney’s definition of chaos, Amer. Math. Monthly, 99, 332-334, (1992) · Zbl 0758.58019
[2] Devaney, R. L., An introduction to chaotic dynamical systems, (1989), Addison-Wesley Reading, MA · Zbl 0695.58002
[3] Glasner, E.; Weiss, B., Sensitive dependence on initial conditions, Nonlinearity, 6, 1067-1075, (1993) · Zbl 0790.58025
[4] Gottschalk, W. H.; Hedlund, G. A., Topological dynamics, Amer. Math. Soc. Colloq. Publ., vol. 36, (1955), Amer. Math. Soc. Providence, RI · Zbl 0067.15204
[5] Guckenheimer, J., Sensitive dependence on initial conditions for one-dimensional maps, Comm. Math. Phys., 70, 133-160, (1979) · Zbl 0429.58012
[6] Huang, W.; Ye, X.-D., Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos, Topology Appl., 117, 259-272, (2002) · Zbl 0997.54061
[7] Kontorovich, E.; Megrelishvili, M., A note on sensitivity of semigroup actions, Semigroup Forum, 76, 133-141, (2008) · Zbl 1161.47029
[8] Li, T.; Yorke, J., Period three implies chaos, Amer. Math. Monthly, 82, 985-992, (1975) · Zbl 0351.92021
[9] Nemytskii, V. V.; Stepanov, V. V., Qualitative theory of differential equations, (1960), Princeton University Press Princeton, NJ · Zbl 0089.29502
[10] Silverman, S., On maps with dense orbits and the definition of chaos, Rocky Mountain J. Math., 22, 353-375, (1992) · Zbl 0758.58024
[11] Walters, P., An introduction to ergodic theory, Grad. Texts in Math., vol. 79, (1982), Springer-Verlag New York, Heidelberg, Berlin · Zbl 0475.28009
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.