• Previous Article
    The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory
  • JMD Home
  • This Issue
  • Next Article
    A dichotomy between discrete and continuous spectrum for a class of special flows over rotations
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
[1]

Valerii Los, Vladimir A. Mikhailets, Aleksandr A. Murach. An isomorphism theorem for parabolic problems in Hörmander spaces and its applications. Communications on Pure & Applied Analysis, 2017, 16 (1) : 69-98. doi: 10.3934/cpaa.2017003

[2]

Eleonora Catsigeras, Yun Zhao. Observable optimal state points of subadditive potentials. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1375-1388. doi: 10.3934/dcds.2013.33.1375

[3]

Dongping Zhuang. Irrational stable commutator length in finitely presented groups. Journal of Modern Dynamics, 2008, 2 (3) : 499-507. doi: 10.3934/jmd.2008.2.499

[4]

Samuel Roth. Constant slope models for finitely generated maps. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2541-2554. doi: 10.3934/dcds.2018106

[5]

André de Carvalho, Toby Hall. Decoration invariants for horseshoe braids. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 863-906. doi: 10.3934/dcds.2010.27.863

[6]

Feng Zhang, Jinting Wang, Bin Liu. Equilibrium joining probabilities in observable queues with general service and setup times. Journal of Industrial & Management Optimization, 2013, 9 (4) : 901-917. doi: 10.3934/jimo.2013.9.901

[7]

Sobhan Seyfaddini. Spectral killers and Poisson bracket invariants. Journal of Modern Dynamics, 2015, 9: 51-66. doi: 10.3934/jmd.2015.9.51

[8]

Rémi Leclercq. Spectral invariants in Lagrangian Floer theory. Journal of Modern Dynamics, 2008, 2 (2) : 249-286. doi: 10.3934/jmd.2008.2.249

[9]

BronisŁaw Jakubczyk, Wojciech Kryński. Vector fields with distributions and invariants of ODEs. Journal of Geometric Mechanics, 2013, 5 (1) : 85-129. doi: 10.3934/jgm.2013.5.85

[10]

Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185

[11]

Konovenko Nadiia, Lychagin Valentin. Möbius invariants in image recognition. Journal of Geometric Mechanics, 2017, 9 (2) : 191-206. doi: 10.3934/jgm.2017008

[12]

A. Yu. Ol'shanskii and M. V. Sapir. Non-amenable finitely presented torsion-by-cyclic groups. Electronic Research Announcements, 2001, 7: 63-71.

[13]

George Papadopoulos, Holger R. Dullin. Semi-global symplectic invariants of the Euler top. Journal of Geometric Mechanics, 2013, 5 (2) : 215-232. doi: 10.3934/jgm.2013.5.215

[14]

Walter D. Neumann and Jun Yang. Invariants from triangulations of hyperbolic 3-manifolds. Electronic Research Announcements, 1995, 1: 72-79.

[15]

Michael C. Sullivan. Invariants of twist-wise flow equivalence. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 475-484. doi: 10.3934/dcds.1998.4.475

[16]

Kingshook Biswas. Complete conjugacy invariants of nonlinearizable holomorphic dynamics. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 847-856. doi: 10.3934/dcds.2010.26.847

[17]

Thomas Fidler, Markus Grasmair, Otmar Scherzer. Identifiability and reconstruction of shapes from integral invariants. Inverse Problems & Imaging, 2008, 2 (3) : 341-354. doi: 10.3934/ipi.2008.2.341

[18]

Paul Loya and Jinsung Park. On gluing formulas for the spectral invariants of Dirac type operators. Electronic Research Announcements, 2005, 11: 1-11.

[19]

John Fogarty. On Noether's bound for polynomial invariants of a finite group. Electronic Research Announcements, 2001, 7: 5-7.

[20]

Michael C. Sullivan. Invariants of twist-wise flow equivalence. Electronic Research Announcements, 1997, 3: 126-130.

2017 Impact Factor: 0.425

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]