On the law of logarithm of the recurrence time

Abstract

Let T be a transformation from I = [0, 1) onto itself and let Q(n) (x) be the subinterval [i/2(n), (i+1)/2(n)), 0 less than or equal to i < 2(n) containing x. Define K-n(x) = min{j ≥ 1 : T-j(x) ∈ Q(n) (x)} and K-n(x,y) = min{j ≥ 1 : Tj-1(y) ∈ Q(n)(x)}. For various transformations defined on I, we show that [GRAPHICS]