Israel Journal of Mathematics

, Volume 113, Issue 1, pp 243–267 | Cite as

On directional entropy functions

  • Kyewon Koh Park
Article
Received:
Revised:

Abstract

Given aZ 2-process, the measure theoretic directional entropy function,h(\(\vec v\)% MathType!End!2!1!), is defined on\(S^1 = \left\{ {\vec v:\left\| {\vec v} \right\| = 1} \right\} \subset R^2 \)% MathType!End!2!1!. We relate the directional entropy of aZ 2-process to itsR 2 suspension. We find a sufficient condition for the continuity of directional entropy function. In particular, this shows that the directional entropy is continuous for aZ 2-action generated by a cellular automaton; this finally answers a question of Milnor [Mil]. We show that the unit vectors whose directional entropy is zero form aG δ subset ofS 1. We study examples to investigate some properties of directional entropy functions.

Keywords

Cellular Automaton Amenable Group Rational Vector Side Boundary Expansive Direction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AR]
    L. M. Abramov and V. A. Rohlin,The entropy of a skew product of measure preserving transformations, American Mathematical Society Translations, Series2 (1965), 255–265.Google Scholar
  2. [BL]
    M. Boyle and D. Lind,Expansive Subdynamics, Transactions of the American Mathematical Society349 (1997), 55–102.MATHCrossRefMathSciNetGoogle Scholar
  3. [He]
    G. A. Hedlund,Endomorphisms and automorphisms of the shift dynamical system, Mathematical Systems Theory3 (1969), 320–375.MATHCrossRefMathSciNetGoogle Scholar
  4. [KP]
    B. Kaminski and K. K. Park,On the directional entropy for Z 2-actions on a Lebesgue space, preprint.Google Scholar
  5. [Li]
    D. Lind, Personal communication.Google Scholar
  6. [Mi1]
    J. Milnor,On the entropy geometry of cellular automata, Complex Systems2 (1988), 357–386.MATHMathSciNetGoogle Scholar
  7. [Mi2]
    J. Milnor,Directional entropies of cellular automaton-maps, NATO Advanced Science Institutes Series F20 (1986), 113–115.MathSciNetGoogle Scholar
  8. [MN]
    B. Marcus and S. Newhouse,Measures of maximal entropy for a class of skew products, Lecture Notes in Mathematics729, Springer-Verlag, Berlin, 1979, pp. 105–125.Google Scholar
  9. [OW1]
    D. Ornstein and B. Weiss,The Shannon-McMillan-Breiman Theorem for a class of amenable groups, Israel Journal of Mathematics44 (1983), 53–60.MATHCrossRefMathSciNetGoogle Scholar
  10. [OW2]
    D. Ornstein and B. Weiss,Entropy and isomorphism theorem for actions of amenable groups, Journal d’Analyse Mathématique48 (1987), 1–141.MATHMathSciNetCrossRefGoogle Scholar
  11. [Pa1]
    K. K. Park,Entropy of a skew product with a Z 2-action, Pacific Journal of Mathematics172 (1995), 227–241.Google Scholar
  12. [Pa2]
    K. K. Park,Continuity of directional entropy, Osaka Journal of Mathematics31 (1994), 613–628.MATHMathSciNetGoogle Scholar
  13. [Pa3]
    K. K. Park,A counterexample of the entropy of a skew product, Indagationes Mathematicae, to appear.Google Scholar
  14. [Ro]
    A. Rosenthal,Uniform generators for ergodic finite entropy free actions of amenable groups, Probability Theory and Related Fields77 (1988), 147–166.MATHCrossRefMathSciNetGoogle Scholar
  15. [Si1]
    Y. Sinai,An answer to a question by J.Milnor, Commentarii Mathematici Helvetici60 (1985), 173–178.MATHCrossRefMathSciNetGoogle Scholar
  16. [Si2]
    Y. Sinai,Topics in Ergodic Theory, Princeton University Press, 1995.Google Scholar
  17. [Th]
    J. P. Thouvenot, Personal communication.Google Scholar

Copyright information

© The Magnes Press 1999

Authors and Affiliations

  • Kyewon Koh Park
    • 1
  1. 1.Department of MathematicsAjou UniversitySuwonKorea

Personalised recommendations