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

• Autorzy: B. Kamiński , K. K. Park
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 28D15: General groups of measure-preserving transformations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1999
• Tom: 133
• Numer: 1
• Strony: 39-51

Daty

wydano

1999

otrzymano

1998-03-12

Twórcy

autor

B. Kamiński

• Faculty of Mathematics and Informatics, Nicholas Copernicus University, Chopina 12/18 87-100 Toruń, Poland

autor

K. K. Park

• Department of Mathematics, College of Natural Sciences, Ajou University, Suwon 441-749, Korea

Bibliografia

• [1] R. M. Belinskaya, Entropy of a piecewise-power skew product, Izv. Vyssh. Uchebn. Zaved. Mat. 1974, no. 3, 12-17 (in Russian).
• [2] M. Boyle and D. Lind, Expansive subdynamics, Trans. Amer. Math. Soc. 349 (1997), 55-102.
• [3] J. P. Conze, Entropie d'un groupe abélien de transformations, Z. Wahrsch. Verw. Gebiete 25 (1972), 11-30.
• [4] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, Berlin, 1982.
• [5] I. Filipowicz, Rank, covering number and the spectral multiplicity function for $ℤ^d$-actions, thesis, Toru/n, 1996.
• [6] 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).
• [7] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. II, Springer, Berlin, 1970.
• [8] 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.
• [9] E. Krug, Folgenentropie für abelsche Gruppen von Automorphismen, thesis, Nürnberg 1973.
• [10] 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.
• [11] J. Milnor, On the entropy geometry of cellular automata, Complex Systems 2 (1988), 357-386.
• [12] D. Newton, On the entropy of certain classes of skew product transformations, Proc. Amer. Math. Soc. 21 (1969), 722-726.
• [13] K. K. Park, Continuity of directional entropy for a class of $ℤ^2$-actions, J. Korean Math. Soc. 32 (1995), 573-582.
• [14] K. K. Park, Entropy of a skew product with a $ℤ^2$-action, Pacific J. Math. 172 (1996), 227-241.
• [15] K. K. Park, A counter-example of the entropy of a skew product, Indag. Math., to appear.
• [16] K. K. Park, On directional entropy functions, Israel J. Math., to appear.
• [17] P. Walters, Some invariant σ-algebras for measure preserving transformations, Trans. Amer. Math. Soc. 163 (1972), 357-368.
• [18] P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982.