A SHRINKING TARGET THEOREM FOR ERGODIC TRANSFORMATIONS OF THE UNIT INTERVAL

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(R_αⁱ 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.