Seminarium Układy dynamiczne

De Finetti's theorem on exchangeable processes states, in its simplest form, that the only Sym(I) invariant ergodic measures on {0,1}^I are the products of identical measures. This means that all Sym(I) invariant random subsets of I can be obtained as follows. First we choose a parameter p\in[0,1] according to some distribution, then we choose a Bernoulli random subset S\subset I, where each element I\in I is contained in S independently with probability p. A natural question presents itself. Do we really need the full Sym(I)-invariance to draw such strong concussions on the structure of invariant random subsets? In a recent joint work with Simon Machado we study random subsets of Z^d, invariant under the group of special affine automorphisms ASL_d(Z).  We prove that these sets arise as Bernoulli percolations with varying parameters, guided by certain random multivariate polynomials modulo 1. To arrive at this description we rely on multiple ergodic averages, concatenation theorems of Tao-Ziegler for characteristic factors and a sprinkle of homogeneous dynamics.