On the directional entropy for ℤ²-actions on a Lebesgue space

B. Kamiński; K. Park

ArticleOriginal scientific text

Title

On the directional entropy for ℤ²-actions on a Lebesgue space

Authors ¹, ²

Affiliations

Faculty of Mathematics and Informatics, Nicholas Copernicus University, Chopina 12/18 87-100 Toruń, Poland
Department of Mathematics, College of Natural Sciences, Ajou University, Suwon 441-749, Korea

Abstract

We define the concept of directional entropy for arbitrary

ℤ^{2}

-actions on a Lebesgue space, we examine its basic properties and consider its behaviour in the class of product actions and rigid actions.

R. M. Belinskaya, Entropy of a piecewise-power skew product, Izv. Vyssh. Uchebn. Zaved. Mat. 1974, no. 3, 12-17 (in Russian).
M. Boyle and D. Lind, Expansive subdynamics, Trans. Amer. Math. Soc. 349 (1997), 55-102.
J. P. Conze, Entropie d'un groupe abélien de transformations, Z. Wahrsch. Verw. Gebiete 25 (1972), 11-30.
I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, Berlin, 1982.
I. Filipowicz, Rank, covering number and the spectral multiplicity function for $ℤ^{d}$ -actions, thesis, Toru/n, 1996.
A. A. Gura, On ergodic dynamical systems with point spectrum and commutative time, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 21 (1966), no. 4, 92-95 (in Russian).
E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. II, Springer, Berlin, 1970.
J. Kieffer, The isomorphism theorem for generalized Bernoulli schemes, in: Studies in Probability and Ergodic Theory, Adv. Math. Suppl. Stud. 2, Academic Press, 1978, 251-267.
E. Krug, Folgenentropie für abelsche Gruppen von Automorphismen, thesis, Nürnberg 1973.
J. Milnor, Directional entropies of cellular automaton-maps, in: Disordered Systems and Biological Organization, NATO Adv. Sci. Inst. Ser. F 20, Springer, 1986, 113-115.
J. Milnor, On the entropy geometry of cellular automata, Complex Systems 2 (1988), 357-386.
D. Newton, On the entropy of certain classes of skew product transformations, Proc. Amer. Math. Soc. 21 (1969), 722-726.
K. K. Park, Continuity of directional entropy for a class of $ℤ^{2}$ -actions, J. Korean Math. Soc. 32 (1995), 573-582.
K. K. Park, Entropy of a skew product with a $ℤ^{2}$ -action, Pacific J. Math. 172 (1996), 227-241.
K. K. Park, A counter-example of the entropy of a skew product, Indag. Math., to appear.
K. K. Park, On directional entropy functions, Israel J. Math., to appear.
P. Walters, Some invariant σ-algebras for measure preserving transformations, Trans. Amer. Math. Soc. 163 (1972), 357-368.
P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982.

Title

On the directional entropy for ℤ²-actions on a Lebesgue space

Affiliations

Abstract

Bibliography