On the existence of invariant absolutely continuous probability measures for C¹ expanding maps of the circle

2023/02/10 by Hamza Ounesli, Ounesli, Hamza
Mathematics · #37A05 #37C40 #37D20 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2302.05339

openalex publication_date 2023/02/10 · openalex created_date 2023/02/15 · openalex updated_date 2026/07/28

Abstract

We prove that for any given modulus of continuity ω there exist (uncountably many) C1 uniformly expanding maps of the circle whose derivatives have C¹ as an optimal modulus of continuity and which preserve an invariant probability measure equivalent to Lebesgue whose density is ω-continuous, and also (uncountably many) C¹ uniformly expanding maps of the circle whose derivatives have ω as an optimal modulus of continuity which preserve Lebesgue measure. Moreover, we show that many of these maps, including those which preserve Lebesgue measure, have unbounded distortion.

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• Non-ergodicity for C¹ Expanding Maps
• Generic expanding maps without absolutely continuous invariant σ-finite measure
• Expanding Measures
• The almost sure invariance principle for unbounded functions of expanding maps
• Invariant measures for typical quadratic maps