Finitary reconstruction of a measure preserving transformation

AbstractThis paper considers thefinitary reconstruction of an ergodic measure preserving transformationT of a complete separable metric spaceX from a single trajectoryx, Tx, …, or more generally, from a suitable reconstruction sequence x=x1,x2, … withxi∈X. Ann-sample reconstruction is a functionTn: Xn+1 →X; the map $$\hat T_n $$ (·;x1, …,xn)is treated as an estimate ofT(·) based on then initial elements of x. Given a reference probability measureμ0 and constantM>1, functionsT1,T2, … are defined, and it is shown that for everyμ with 1/M≤dμ/dμ0≤M, everyμ-preserving transformationT, and every reconstruction sequence x forT, the estimates $$\hat T_n $$ (·;x1, …,xnconverge toT in the weak topology.For the family of interval exchange transformations of [0, 1] a simple family of estimates is described and shown to be consistent both pointwise and in the strong topology. However, it is also shown that no finitary estimation scheme is consistent in the strong topology for the family of all ergodic Lebesgue measure preserving transformations of the unit interval, even if x is assumed to be a generic trajectory ofT.