A universal law of logarithm of the recurrence time

Abstract

A point x in [0, 1] is represented as a binary expansion, i.e. it is identified with an infinite binary sequence of 0 and I. Given a map T satisfying 0 less than or equal to T (x) less than or equal to 1 for 0 less than or equal to x less than or equal to 1, we iterate the map T until the first n bits in x recur as the first n bits in the K(n)th iterate T-Kn(x) for some K-n = K-n (x). We call K-n (x) the nth recurrence time of x. More precisely, put E-n,E-j = [ (j - 1)/2(n), j/2(n)), 1 less than or equal to j less than or equal to 2(n), and let E-n(x) be one of the intervals E-n,E-j containing x. Then K-n(x) = min{j greater than or equal to 1 : T-j(x) is an element of E-n(x)}. For higher dimensional cases we define the recurrence time using subcubes instead of subintervals. For a wide class of T including Henon mappings we present two conjectures: first, if T is ergodic and has positive entropy, then the sequence of averages of (log(2) K-n)/n monotonically converges to the Hausdorff dimension as n --> infinity. Second, the values of KnPn are exponentially distributed as n --> infinity where P-n(x) is the measure of E-n(x). To support our conjectures computer simulations are presented.