Failure of the Pointwise Ergodic Theorem - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T21:48:53Z http://mathoverflow.net/feeds/question/118978 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118978/failure-of-the-pointwise-ergodic-theorem Failure of the Pointwise Ergodic Theorem Joel Moreira 2013-01-15T15:08:07Z 2013-01-15T15:15:46Z <p>It is known that Birkhoff's pointwise ergodic theorem (unlike von Neumann's mean ergodic Theorem) fails to hold for general Folner sequences. The counter-example usually given is the Folner sequence <code>\$F_N=\{N^2,N^2+1...,N^2+N\}\$</code>, however on the references (including the first such result by Akcoglu and del Junco in 1975) the Folner sequence which fails is actually <code>\$G_N=\{N,N+1,...,N+\lfloor N\rfloor\}\$</code>. Since <code>\$\{F_N\}\$</code> is just a (quite sparse) subsequence of <code>\$\{G_N\}\$</code> it could happen that the ergodic averages converge pointwise along <code>\$\{F_N\}\$</code> even if they don't along <code>\$\{G_N\}\$</code>.</p> <p>Moreover, Lindenstrauss showed that every Folner sequence has a subsequence along which convergence holds.</p> <p>So I'm just curious if there is a published proof (or if it is easy and I'm just overlooking something) that the sequence <code>\$\{F_N\}\$</code> is not good for pointwise ergodic theorem.</p> http://mathoverflow.net/questions/118978/failure-of-the-pointwise-ergodic-theorem/118980#118980 Answer by Anthony Quas for Failure of the Pointwise Ergodic Theorem Anthony Quas 2013-01-15T15:15:46Z 2013-01-15T15:15:46Z <p>Bellow, Alexandra(1-NW); Jones, Roger(1-DPL); Rosenblatt, Joseph(1-OHS) Convergence for moving averages. Ergodic Theory Dynam. Systems 10 (1990), no. 1, 43–62. </p>