zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Measures of chaos and a spectral decomposition of dynamical systems on the interval. (English) Zbl 0812.58062

Various notions of chaos of dynamical systems on the interval have been proposed by different authors. Among them there is the definition of chaotic functions given by Li and Yorke and another notion based on the topological entropy. In this paper the authors use the upper and lower distance distribution functions associated to a continuous function on the interval to measure the degree of chaos of the function.

Let f be a continuous function on the interval. For x,y[0,1], the upper and lower distribution functions, F xy * and F xy , are defined for any t0 as the lim sup and lim inf as n of the average number of times that the distance |f i (x)-f i (y)| between the trajectories of x and y is less than t during the first n iterations. Both functions are nondecreasing and may be viewed as cumulative probability distribution functions.

Among the set of all lower distribution functions associated to a given f, the authors pick out a subset, Σ(f), called the spectrum of f. The definition of the spectrum is based on initial results on dynamical systems concerning the notion of scrambled set and periodic decompositions. Thanks to a very simple example the authors are able to illustrate the ideas behind the use of this definition and to measure chaos.

The main results relate previous notions of chaos with the one proposed here: 1) If f has positive topological entropy, then the spectrum is nonempty and finite, and any element of the spectrum is zero on an interval [0,ε] (thus relating the spectrum with the notion of chaos of Li and Yorke). 2) If f has zero topological entropy, then the spectrum is just the constant unit function.

37D45Strange attractors, chaotic dynamics
37A30Ergodic theorems, spectral theory, Markov operators
54C70Topological entropy
26A18Iteration of functions of one real variable
37B99Topological dynamics