Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1071.37012
Banks, John
Chaos for induced hyperspace maps. (English)
[J] Chaos Solitons Fractals 25, No. 3, 681-685 (2005). ISSN 0960-0779

Summary: For $(X, d)$ be a metric space, $f: X\to X$ a continuous map and $({\Cal K}(X),H)$ the space of nonempty compact subsets of $X$ with the Hausdorff metric, one may study the dynamical properties of the induced map $$\overline f:{\Cal K}(X)\to{\Cal K}(X): A\mapsto f(A).$$ {\it H. Román-Flores} [Chaos Solitons Fractals 17, 99--104 (2003; Zbl 1098.37008)] has shown that if $f$ is topologically transitive then so is $f$, but that the reverse implication does not hold. This paper shows that the topological transitivity of $\overline f$ is in fact equivalent to weak topological mixing on the part of $f$. This is proved in the more general context of an induced map on some suitable hyperspace ${\Cal H}$ of $X$ with the Vietoris topology which agrees with the topology of the Hausdorff metric in the case discussed by Roman-Flores.

MSC 2000:: *37B99 Topological dynamics
37A25 Ergodicity, mixing, rates of mixing
37B05 Transformations and group actions with special properties
54H20 Topological dynamics

Keywords: sensitive dependence on initial conditions; induced maps; hyperspaces; periodic pints; topological transitivity; weak topological mixing

Citations: Zbl 1098.37008

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]