### The law of series

Yves Lacroix

Abstract.
The law of series in ergodic theory comes out as an
interpretation of a surprising phenomenon appearing in ergodic
processes presenting positive entropy, and discovered in 2006. This
phenomenon is connected to the theory of asymptotics of return times
to long cylinder sets in ergodic symbolic processes, that is weak
convergence of normalized distribution functions for return times to
typical cylinder sets.
For fast enough mixing systems sevral studies have proven that mixing
grants that the distributionnal behaviour of entry or return times to
those cylinders is approximately exponential with parameter 1, but not
much was known earlier about other possible behaviours in generic
processes, or arbitrary processes presenting a chaotic behaviour in
some sense (positive entropy).
We came to prove that for chaotic systems the asymptotic distribution
for entry times cannot exceed the exponential law with parameter 1,
and moreover that typically for such a process a partition generates a
symbolic factor for which the distribution is ultimately degenerate
(equals z?ro in fact !). This extends to zero entropy processes and
also supports approximate returns in a d-bar metric sense.
A simple argument using preservation of measure shows how this
connects to the appearence of clusters of rare events, which somehow
gives a first interpretation for the law of series in this very
general and simple mathematical model, contrary to the interpretation
of Warren Weaver.
Joint Works (2002-2011) with T. Downarowicz, N. Haydn, P. Grzegorek,
M. Kupsa, D. Léandri, S. Vaienti.