We define the concept of directional entropy for arbitrary -actions on a Lebesgue space, we examine its basic properties and consider its behaviour in the class of product actions and rigid actions.
@article{bwmeta1.element.bwnjournal-article-smv133i1p39bwm, author = {B. Kami\'nski and K. Park}, title = {On the directional entropy for $\mathbb{Z}$$^2$-actions on a Lebesgue space}, journal = {Studia Mathematica}, volume = {133}, year = {1999}, pages = {39-51}, zbl = {0932.28015}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv133i1p39bwm} }
Kamiński, B.; Park, K. On the directional entropy for ℤ²-actions on a Lebesgue space. Studia Mathematica, Tome 133 (1999) pp. 39-51. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv133i1p39bwm/
[00000] [1] R. M. Belinskaya, Entropy of a piecewise-power skew product, Izv. Vyssh. Uchebn. Zaved. Mat. 1974, no. 3, 12-17 (in Russian).
[00001] [2] M. Boyle and D. Lind, Expansive subdynamics, Trans. Amer. Math. Soc. 349 (1997), 55-102. | Zbl 0863.54034
[00002] [3] J. P. Conze, Entropie d'un groupe abélien de transformations, Z. Wahrsch. Verw. Gebiete 25 (1972), 11-30. | Zbl 0261.28015
[00003] [4] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, Berlin, 1982.
[00004] [5] I. Filipowicz, Rank, covering number and the spectral multiplicity function for -actions, thesis, Toru/n, 1996.
[00005] [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).
[00006] [7] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. II, Springer, Berlin, 1970. | Zbl 0213.40103
[00007] [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.
[00008] [9] E. Krug, Folgenentropie für abelsche Gruppen von Automorphismen, thesis, Nürnberg 1973.
[00009] [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. | Zbl 1330.37015
[00010] [11] J. Milnor, On the entropy geometry of cellular automata, Complex Systems 2 (1988), 357-386. | Zbl 0672.68025
[00011] [12] D. Newton, On the entropy of certain classes of skew product transformations, Proc. Amer. Math. Soc. 21 (1969), 722-726. | Zbl 0174.09201
[00012] [13] K. K. Park, Continuity of directional entropy for a class of -actions, J. Korean Math. Soc. 32 (1995), 573-582. | Zbl 0847.28007
[00013] [14] K. K. Park, Entropy of a skew product with a -action, Pacific J. Math. 172 (1996), 227-241. | Zbl 0868.28011
[00014] [15] K. K. Park, A counter-example of the entropy of a skew product, Indag. Math., to appear.
[00015] [16] K. K. Park, On directional entropy functions, Israel J. Math., to appear. | Zbl 0937.37003
[00016] [17] P. Walters, Some invariant σ-algebras for measure preserving transformations, Trans. Amer. Math. Soc. 163 (1972), 357-368. | Zbl 0227.28011
[00017] [18] P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982. | Zbl 0475.28009