Asymptotic likelihood of chaos for smooth families of circle maps

2009/08/04 by Hiroki Takahasi, Takahasi, Hiroki
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS

paper · pdf · doi:10.48550/arxiv.0908.0448

22 pages

arxiv created 2009/08/04 · arxiv updated 2009/12/01

Abstract

We consider a smooth two-parameter family f_a,L\colonθ↦ θ+a+LΦ(θ) of circle maps with a finite number of critical points. For sufficiently large L we construct a set A_L^(∞) of a-values of positive Lebesgue measure for which the corresponding f_a,L exhibits an exponential growth of derivatives along the orbits of the critical points. Our construction considerably improves the previous one of Wang and Young for the same class of families, in that the following asymptotic estimate holds: the Lebesgue measure of A_L^(∞) tends to full measure in a-space as L tends to infinity.

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