Journal of Modern Dynamics (JMD)

Entropy is the only finitely observable invariant

Pages: 93 - 105, Issue 1, January 2007      doi:10.3934/jmd.2007.1.93

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Donald Ornstein - Department of Mathematics, Stanford University, Stanford, CA 94305, United States (email)
Benjamin Weiss - Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel (email)

Abstract: Our main purpose is to present a surprising new characterization of the Shannon entropy of stationary ergodic processes. We will use two basic concepts: isomorphism of stationary processes and a notion of finite observability, and we will see how one is led, inevitably, to Shannon's entropy. A function $J$ with values in some metric space, defined on all finite-valued, stationary, ergodic processes is said to be finitely observable (FO) if there is a sequence of functions $S_{n}(x_{1},x_{2},...,x_{n})$ that for all processes $\mathcal{X}$ converges to $J(\mathcal{X})$ for almost every realization $x_{1}^{\infty}$ of $\mathcal{X}$. It is called an invariant if it returns the same value for isomorphic processes. We show that any finitely observable invariant is necessarily a continuous function of the entropy. Several extensions of this result will also be given.

Keywords:  entropy, finitely observable, isomorphism invariants, finitary isomorphism.
Mathematics Subject Classification:  Primary 37A35, Secondary 60G10.

Received: May 2006;      Available Online: October 2006.