Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

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

Highlights
Master Server