Variation-norm and fluctuation estimates for ergodic bilinear averages

Yen Do, Richard Oberlin, Eyvindur A. Pallson

For any dynamical system, we show that higher variation-norms for
the sequence of ergodic bilinear averages of two functions satisfy a
large range of bilinear *L ^{p}* estimates. It follows
that, with probability one, the number of fluctuations along this
sequence may grow at most polynomially with respect to (the growth of)
the underlying scale. These results strengthen previous works of Lacey
and Bourgain where almost surely convergence of the sequence was
proved (which is equivalent to the qualitative statement that the
number of fluctuations is finite at each scale). Via transference, the
proof reduces to establishing new bilinear