From: Elisha Kobre <ekobre@erols.com>
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