Expansion properties of double standard maps

2022/07/05 by MICHAEL BENEDICKS, Michael Benedicks, Michał Misiurewicz +3 · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics

paper · doi:10.1017/etds.2022.45

openalex publication_date 2022/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

Abstract For the family of double standard maps f_a,b=2x+a+(b/π ) sin 2π x \pmod 1 we investigate the structure of the space of parameters a when b=1 and when b∈ [0,1) . In the first case the maps have a critical point, but for a set of parameters E₁ of positive Lebesgue measure there is an invariant absolutely continuous measure for f_a,1 . In the second case there is an open non-empty set E_b of parameters for which the map f_a,b is expanding. We show that as b\nearrow 1 , the set E_b accumulates on many points of E₁ in a regular way from the measure point of view.

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