On processes which cannot be distinguished by finitary observation

Abstract

A function $J$ defined on a family $C$ of stationary processes is finitely observable if there is a sequence of functions $s_n$ such that $s_n(x_1 ... x_n)\to J(X)$ in probability for every process $X=(x_n)\in C$. Recently, Ornstein and Weiss roved the striking result that if $C$ is the class of aperiodic ergodic finite valued processes, then the only finitely observable isomorphism invariant on $C$ is entropy. We sharpen this in several ways. Our main theorem is that if $X \to Y$ is a zero-entropy extension of finite entropy ergodic systems and $C$ is the family of processes arising from $X$ and $Y$, then every finitely observable function on $C$ is constant. This implies Ornstein and Weiss' result, and extends it to many other families of processes, e.g. it shows that there are no nontrivial finitely observable isomorphism invariants for processes arising from Kronecker systems, mild and strong mixing zero entropy systems. It also implies that any finitely observable isomorphism invariant defined on the family of processes arising from irrational rotations must be constant for rotations belonging to a set of full Lebesgue measure.