  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 Lp 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 Lp bounds for variation-norms of truncated bilinear
operators on R, and the main ingredient of the proof of these bounds is a
variation-norm extension of maximal Bessel inequalities of Lacey and
Demeter--Tao--Thiele.
