zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Chaos for induced hyperspace maps. (English) Zbl 1071.37012

Summary: For (X,d) be a metric space, f:XX a continuous map and (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

f ¯:𝒦(X)𝒦(X):Af(A)·

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 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 of X with the Vietoris topology which agrees with the topology of the Hausdorff metric in the case discussed by Roman-Flores.

MSC:
37B99Topological dynamics
37A25Ergodicity, mixing, rates of mixing
37B05Transformations and group actions with special properties
54H20Topological dynamics