A SHRINKING TARGET THEOREM FOR ERGODIC TRANSFORMATIONS OF THE UNIT INTERVAL

Shrey Sanadhya

Research output: Contribution to journalArticlepeer-review

Abstract

We show that for any ergodic Lebesgue measure preserving transformation f : [0, 1) → [0, 1) and any decreasing sequence {bi}i=1 of positive real numbers with divergent sum, the set (equation presented) ∞ ∞ ∩ ∪ f-i(B(Rαi x, bi)) n=1 i=n 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 Rα : [0, 1) → [0, 1) is rotation by α ∈ [0, 1). As a corollary, we provide partial answer to a question asked by Chaika (Question 3, [2]) in the context of interval exchange transformations.

Original languageEnglish
Pages (from-to)4003-4011
Number of pages9
JournalDiscrete and Continuous Dynamical Systems
Volume42
Issue number8
DOIs
StatePublished - 1 Aug 2022

Keywords

  • Shrinking target problem
  • interval exchange transformations

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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