The shift action on independent random variables
The Bernoulli shift model which is the action on a sequence of i.i.d. random variables by time shifts in one of the central examples of classical ergodic theory. To this date much is known on the ergodic theoretic properties of this model. A notable example is Sinai factor theorem which says that if a given system has positive entropy (is chaotic) then it has Bernoulli shift models as factors (subsystems) which leads to one of the most common definitions of chaos. In addition, Ornstein theorem says that two Bernoulli systems are isomorphic if and only if they have the same measure theoretic entropy while Dye's theorem implies that all Bernoulli shifts are orbit equivalent.
Relaxing the i.i.d. hypothesis to independence one arrives to a very natural model of Bernoulli shifts which are not at equilibrium. Up to recent years very little was known on the ergodic theoretic properties of these important systems and in the recent decade a beautiful theory emerged with connections and applications to probability theory (exchangeability), smooth dynamics (new classes of Anosov diffeomorphisms) and geometric group theory ($l^2$ Betti numbers, Stallings ends theorem). I will survey (some of) these results and some of the future challenges. Based on several works, among them is a joint work with Michael Bjorklund and Stefaan Vaes and two joint works with Terry Soo.
תאריך עדכון אחרון : 04/03/2021