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January  2007, 1(1): 93-105. doi: 10.3934/jmd.2007.1.93

## Entropy is the only finitely observable invariant

 1 Department of Mathematics, Stanford University, Stanford, CA 94305, United States 2 Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

Received  May 2006 Published  October 2006

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.
Citation: Donald Ornstein, Benjamin Weiss. Entropy is the only finitely observable invariant. Journal of Modern Dynamics, 2007, 1 (1) : 93-105. doi: 10.3934/jmd.2007.1.93
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