A shrinking target theorem for ergodic transformations of the unit interval

2021/05/01 by Shrey Sanadhya, Sanadhya, Shrey
Computer Science · Mathematics · #37A05 #37A10 #37E10 #Advanced Mathematical Modeling in Engineering #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2105.00301

openalex publication_date 2021/05/01 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We show that for any ergodic Lebesgue measure preserving transformation f: [0,1) → [0,1) and any decreasing sequence \b_i\_i=1^∞ of positive real numbers with divergent sum, the set \undersetn=1\overset∞∩ \underseti=n\overset∞∪ f^-i(B (R_αⁱ x,b_i)) has full Lebesgue measure for almost every x ∈ [0,1) and almost every α∈ [0,1). Here B(x,r) is the ball of radius r centered at x ∈ [0,1) and R_α: [0,1) → [0,1) is rotation by α∈ [0,1). As a corollary, we provide partial answer to a question asked by Chaika in the context of interval exchange transformations.

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