Unfolding of chaotic unimodal maps and the parameter dependence of natural measures
2001 (English)In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 14, no 2, 323-337 p.Article in journal (Refereed) Published
We consider one-parameter families f(a) of interval maps, and discuss the structure in parameter space and the (dis)continuity properties of the natural measure as a function of the parameter near certain strongly chaotic maps (post-critically finite Misiurewicz maps and Benedicks-Carleson maps). In particular, it is shown that the mapping a bar right arrow mu (a) (the natural measure of f(a)) is severely discontinuous at these strongly cli8dtii: maps and is not continuous on any full measure set of parameters in full, generic families. Going in the other direction, it is also shown that if such a chaotic map has a measure for which the critical point is generic, then this measure can be,approximated with measures supported on periodic attractors of nearby maps. The main idea is to construct cascades of post-critically finite Misiurewicz map and cascades of maps with periodic attractors, whose critical oibits reproduce various invariant sets of the unperturbed map. In the special case of the quadratic family, generalizations can be made to any non-renormalizable maps.
Place, publisher, year, edition, pages
2001. Vol. 14, no 2, 323-337 p.
continuous invariant-measures, piecewise monotonic transformations, one-dimensional maps, quadratic family, generic properties, dynamics
IdentifiersURN: urn:nbn:se:kth:diva-20500ISI: 000167838900008OAI: oai:DiVA.org:kth-20500DiVA: diva2:339195
QC 201005252010-08-102010-08-10Bibliographically approved