Superpolynomial growth in the number of v_n names for random walks on random sceneries

Abstract

For any stationary process X=X_1,X_2,\dotsc with shift map \sigma, and T an invertible, measure-preserving flow on a probability space, consider the random walk on a random scenery \hat{T}_X(x,\omega)=(\sigma(x),T_{x_0}(\omega)). We prove that if X satisfies a certain property and T has positive entropy, then the number of names in the Vershik metric v_n increases superpolynomially in n. This fact allows us to prove that a class of smooth maps are not standard in the sense of Vershik.