We analyze certain parametrized families of one-dimensional maps with infinitely many critical points from the measure-theoretical point of view. We prove that such families have absolutely continuous invariant probability measures for a positive Lebesgue measure subset of parameters. Moreover, we show that both the density of such a measure and its entropy vary continuously with the parameter. In addition, we obtain exponential rate of mixing for these measures and also show that they satisfy the Central Limit Theorem.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm203-3-2,
author = {V\'\i tor Ara\'ujo and Maria Jos\'e Pacifico},
title = {Physical measures for infinite-modal maps},
journal = {Fundamenta Mathematicae},
volume = {205},
year = {2009},
pages = {211-262},
zbl = {1173.37015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm203-3-2}
}
Vítor Araújo; Maria José Pacifico. Physical measures for infinite-modal maps. Fundamenta Mathematicae, Tome 205 (2009) pp. 211-262. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm203-3-2/