Commuting measure-preserving transformations
Tóm tắt
Let φ1, ... ,φ
d
be commuting measure-preserving transformations,
$$
\phi ^l \equiv \phi _1^{l_1 } \phi _2^{l_2 } \cdot \cdot \cdot \phi _d^{l_d } ,\Phi = \left\{ {\phi ^l } \right\}
$$
. The Kakutani-Rokhlin tower theorem is proved in a refined form for non-periodic groups Φ, and the Shannon-McMillan theorem is extended to ergodic groups. These results are used to extend recent isomorphism results to groups of transformations.
Tài liệu tham khảo
N. Dunford and J. T. Schwartz,Linear Operators I, (Chapter 8), N. Y., 1958.
P. Halmos,Lectures on Ergodic Theory, The Mathematical Society of Japan, Tokyo, 1956.
A. A. Kirillov,Dynamical systems, factors and group representations, Uspehi Mat. Nauk22 (1967), 67–80.
W. Krieger,On entropy and generators of measure-preserving transformations, TAMS149 (1970), 453–464.
D. S. Ornstein,Bernoulli shifts with the same entropy are isomorphic, Advances in Math.4 (1970), 337–352.
D. S. Ornstein,Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Math.6 (1970), 339–348.
D. S. Ornstein,Factors of Bernoulli shifts are Bernoulli shifts, Advances in Math.5 (1970), 349–364.
V. A. Rokhlin,A general measure-preserving transformation is not mixing, Dokl. Akad. Nauk SSSR60 (1948), 349–351.
V. A. Rokhlin,Lectures on the entropy theory of transformations with invariant measure, Uspehi Math. Nauk22 (1967), 3–56.
D. Ruelle,Statistical Mechanics, N.Y., 1969.
M. Smorodinsky,Ergodic Theory, Entropy, Lecture Notes in Mathematics, No. 214, 1971.
B. Weiss,The isomorphism problem in ergodic theory, BAMS (to appear).