A Chaotic Continuous Map Generates All Probability Distributions

✍ Scribed by F. Balibrea; J. Smital

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 439 KB
• Volume: 180
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.1993.1418

✦ Synopsis

Let (f) be a continuous map of the compact unit interval (l=[0,1]), such that (f^{2}), the second iterate of (f), is topologically transitive in (I). If for some (x) and (y) in (I) and any (t) in (I) there exists (\lim (1 / n) #\left{i \leqslant n ;\left|f^{i}(x)-f^{i}(y)\right|<t\right}) for (n \rightarrow \infty), denote it by (\varphi_{x y}(t)). In the paper we consider the class (\mathscr{F}(f)) if all (\varphi_{x y}). The main results are that (\mathscr{F}(f)) is convex and pointwise closed. Using this we show that (\mathscr{F}(f)) is always bigger than the class (\mathscr{O}(f)) of probability distributions generated analogously by single trajectories (and corresponding to the class of probability invariant measures of (f) ), and prove that there are universal generators of probability distributions, i.e., maps (f) such that (\bar{F}(f)) is the class (\mathscr{M}) of all non-decreasing functions (I \rightarrow I) (contrary to this, (\mathscr{A}(f) \supset \mathscr{M}) for no (f) ). These results can be extended to more general continuous maps. One of the possible applications is to use the size of (\mathscr{F}(f)) as a measure of the degree of chaos of (f). 1993 Academic Press, Inc