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Non-standard smooth realizations of Liouville rotations

Published online by Cambridge University Press:  01 December 2007

B. R. FAYAD
Affiliation:
LAGA, UMR 7539, Université Paris 13, 93430 Villetaneuse, France (email: fayadb@math.univ-paris13.fr)
M. SAPRYKINA
Affiliation:
Department of Mathematics and Statistics, Jeffery Hall, University Avenue, Kingston, ON Canada, K7L 3N6 (email: masha@mast.queensu.ca)
A. WINDSOR
Affiliation:
Department of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712-0257, USA (email: awindsor@math.utexas.edu)

Abstract

We augment the conjugation approximation method with explicit estimates on the conjugacy map. This allows us to construct ergodic volume-preserving diffeomorphisms measure-theoretically isomorphic to any a priori given Liouville rotation on a variety of manifolds. In the special case of tori the maps can be made uniquely ergodic.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Anosov, D. V. and Katok, A. B.. New examples in smooth ergodic theory. Ergodic diffeomorphisms. Tr. Mosk. Mat. Ob. 23 (1970), 336.Google Scholar
[2]Fayad, B. and Katok, A.. Constructions in elliptic dynamics. Ergod. Th. & Dynam. Sys. 24(5) (2004), 14771520.Google Scholar
[3]Fayad, B. and Saprykina, M.. Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary. Ann. Sci. École Norm. Sup. (4) 38(3) (2005), 339364.Google Scholar
[4]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995.Google Scholar
[5]Saprykina, M.. Analytic nonlinearizable uniquely ergodic diffeomorphisms on . Ergod. Th. & Dynam. Sys. 23(3) (2003), 935955.Google Scholar
[6]Windsor, A.. Minimal but not uniquely ergodic diffeomorphisms. Smooth Ergodic Theory and its Applications (Seattle, WA, 1999) (Proceedings Symposia in Pure Mathematics, 69). American Mathematical Society, Providence, RI, 2001, pp. 809824.Google Scholar