×

zbMATH — the first resource for mathematics

Singular effects in chaotic dynamical systems. (English. Russian original) Zbl 0820.58039
Russ. Acad. Sci., Dokl., Math. 47, No. 1, 1-5 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 328, No. 1, 7-10 (1993).
At the present time a rather complete picture of the behavior of chaotic dynamical systems has been put together. One of the central places in the theory of such systems is occupied by dynamical systems whose time- dynamics is defined by hyperbolic or (in the nonsmooth case) by piecewise expanding (p.e.) mappings. There are many theoretical results concerning the ergodic properties of similar systems, such as the existence of invariant measures absolutely continuous (a.c.) with respect to Lebesgue measure on unstable leaves and the mixing properties of such measures: exponential rate of decay of correlations, a central limit theorem, stability with respect to small deterministic and random perturbations, etc., proved under the assumption of a number of technical conditions. However, computer simulation of such systems usually does not show any differences between systems in which these conditions hold and systems in which they do not hold. For example, all the properties mentioned above are usually valid for one-dimensional p.e. mappings with positive Lyapunov exponent.
The aim of the present note is to discuss the question of the necessity of these conditions and to construct a number of simple examples of mappings for which the absence of some these conditions leads to a localization effect – the occurrence of new ergodic components that are concentrated on sets of small measure.

MSC:
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37A99 Ergodic theory
28D05 Measure-preserving transformations
37D99 Dynamical systems with hyperbolic behavior
PDF BibTeX XML Cite