2009/11/28 by Daniel Schnellmann, Schnellmann, Daniel
Mathematics · #Advanced Topology and Set Theory #Mathematical Dynamics and Fractals #math.DS #msc:37A05 #msc:37D20 #msc:37E05
paper · pdf · doi:10.48550/arxiv.0911.5411
33 pages, 3 figures; inclusion of a new section about almost sure typicality in transversal families of piecewise expanding unimodal maps; in the first part of the paper the conditions in order to obtain almost sure typicality are weakened; several other (small) improvements
arxiv created 2011/07/17 · arxiv updated 2011/07/19
Let I⊂ℝ be an interval and Ta:[0,1]→[0,1], a∈ I, a one-parameter family of piecewise expanding maps such that for each a∈ I the map Ta admits a unique absolutely continuous invariant probability measure μa. We establish sufficient conditions on such a one-parameter family such that a given point x∈[0,1] is typical for μa for a full Lebesgue measure set of parameters a, i.e. (1)/(n)∑i=0n-1δTai(x) \oversetweak-*\longrightarrowμa,\qquadas n→∞, for Lebesgue almost every a∈ I. In particular, we consider C1,1(L)-versions of β-transformations, skew tent maps, and Markov structure preserving one-parameter families. For the skew tent maps we show that the turning point is almost surely typical.