Math Seminars
Omri Sarig
(PSU)
What to do when the pointwise ergodic theorem doesn't work
The ergodic theorem never works for ergodic infinite measure preserving systems: if one divides the ergodic sums f+f(Tx)+...+f(T^n x) by too much - then they converge to zero, if by too little - then they diverge to infinity, and if by exactly the right amount - then they oscillate without converging (Aaronson).
But sometimes, one can obtain almost sure pointwise convergence by appealing to a suitable summability method. In the case of certain horocycle flows, weighted cesaro summability is sufficient. This is joint work with F. Ledrappier.