From: Elisha Kobre Subject: Borel-Cantelli lemma Date: Sun, 10 Oct 1999 16:37:34 -0400 Newsgroups: sci.math Hi, I recently proved the Borel-Cantelli lemma (It states that if T is a measure preseving transformation on a measure space (X,B,m) and if we have a sequence of sets A_n in B such that Sum(m(A_n)) is finite then for a.e. x there is an N=N(x) such that T^(n)(x) is not in A_n for all n>N.) Now I found an exercise that says to use this lemma to show that if f is a measure preserving map on the 2-Torus and G is curve of finite length on the Torus then for a.e. x, 1/n log d( f^(n)(x), G) goes to 0 as n goes to infinity. (d is a metric) Any Ideas ? Thanks ek