On processes which cannot be distinguished by finite observation

Yonatan Gutman*, Michael Hochman

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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 ) → J(X) in probability for every process X=(x n ) C. Recently, Ornstein and Weiss proved the striking result that if C is the class of aperiodic ergodic finite valued processes, then the only finitely observable isomorphism invariant defined on C is entropy [8]. We sharpen this in several ways. Our main result is that if X → Y is a zero-entropy extension of finite entropy ergodic systems and C is the family of processes arising from generating partitions of 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 follows that there are no nontrivial finitely observable isomorphism invariants for processes arising from the class of Kronecker systems, the class of mild mixing zero entropy systems, or the class of strong mixing zero entropy systems. It also follows that for the class of processes arising from irrational rotations, every finitely observable isomorphism invariant must be constant for rotations belonging to a set of full Lebesgue measure.

Original languageAmerican English
Pages (from-to)265-284
Number of pages20
JournalIsrael Journal of Mathematics
StatePublished - Mar 2008
Externally publishedYes

Bibliographical note

Funding Information:
∗ This research was supported by the Israel Science Foundation (grant No. Received August 13, 2006 and in revised form November 5, 2006.


Dive into the research topics of 'On processes which cannot be distinguished by finite observation'. Together they form a unique fingerprint.

Cite this