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Entropy for smooth abelian actions

Author: David Fried
Journal: Proc. Amer. Math. Soc. 87 (1983), 111-116
MSC: Primary 54H20; Secondary 28D20, 57S15, 58F11
MathSciNet review: 677244
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Abstract: We generalize the usual notions of metric and topological entropy for flows to actions of compactly generated abelian Lie groups. Unlike previous generalizations, ours is nontrivial for smooth actions of $ {{\mathbf{R}}^n}$, $ n \geqslant 2$. We prove some elementary properties of our definitions and we relate them to characteristic exponents and the entropy conjecture.

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  • [AKM] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319. MR 0175106 (30:5291)
  • [B] R. Bowen, Entropy and the fundamental group, Lecture Notes in Math., vol 668, Springer-Verlag, New York, 1978. MR 518545 (80d:58049)
  • [C] J. P. Conze, Entropie d'un groupe abelian de transformations, Z. Wahrsch. Verw. Gebiete 25 (1972), 11-30. MR 0335754 (49:534)
  • [FS] D. Fried and M. Shub, Entropy, linearity and chain recurrence, Inst. Hautes Etudes Sci. Publ. Math. 50 (1980), 203-214. MR 556587 (81a:58033)
  • [KW] Y. Katznelson and B. Weiss, Commuting measure preserving transformations, Israel J. Math. 12 (1972), 161-173. MR 0316680 (47:5227)
  • [K] A. A. Kirillov, Dynamical systems, factors and group representations, Russian Math Surveys 22 (1967). MR 0217256 (36:347)
  • [MP] M. Misiurewicz and F. Przytycki, The entropy conjecture on tori, preprint. MR 0458502 (56:16702)
  • [O] V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968), 197-231. MR 0240280 (39:1629)
  • [P] J. B. Pesin, Lyapunov characteristic exponents and smooth ergodic theory, Russian Math. Surveys 32 (1977), 55-114. MR 0466791 (57:6667)
  • [R] D. Ruelle, Thermodynamic formalism, Addison-Wesley, Reading, Mass., 1978. MR 511655 (80g:82017)
  • [S] M. Shub, Some dynamics of pseudo-Anosov diffeomorphisms, Astérisque 66-67 (1979), 181 -208.
  • [T] J. P. Thouvenot, Convergence en moyenne de l'information pour l'action de $ {{\mathbf{Z}}^2}$, Z. Wahrsch. Verw. Gebiete 24 (1972), 135-137. MR 0321612 (47:10145)

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Article copyright: © Copyright 1983 American Mathematical Society

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