Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-25T13:14:32.641Z Has data issue: false hasContentIssue false
  • English
  • Français

Isentropic fitting of Anosov automorphisms

Published online by Cambridge University Press:  19 September 2008

David Fried
David Fried
Affiliation:
Department of Mathematics, University of California, Santa Cruz, California 95064
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In response to a question of Newhouse, we show that in many cases an Anosov automorphism can be uniformly approximated by a Smale diffeomorphism of the same topological entropy. We conjecture this can be done whenever the automorphism is sufficiently hyperbolic.

A three-dimensional example is given that suggests this approximation is not always possible.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 2 , Issue 2 , June 1982 , pp. 173 - 183
Copyright
Copyright © Cambridge University Press 1982

References

REFERENCES

[1]Bowen, R.. One-dimensional hyperbolic sets for flows. J. Diff. Eqns. 12 (1972), 173–179.CrossRefGoogle Scholar
[2]Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 377–397.CrossRefGoogle Scholar
[3]Denker, M., Grillenberger, C. & Sigmund, K.. Ergodic Theory on Compact Spaces. Lecture Notes in Math. No. 527. Springer: Berlin, 1976.CrossRefGoogle Scholar
[4]Franks, J.. Anosov diffeomorphisms. Global Analysis XIV (Proc. Symp. Pure Math. 1968), pp. 61–93. Amer. Math. Soc: Princeton, 1968.Google Scholar
[5]Franks, J.. Homology and Dynamical Systems. Notes from conf. at Emory University 1980.Google Scholar
[6]Fried, D.. Efficiency vs. hyperbolicity on tori. Global Theory of Dynamic Systems. Lecture Notes in Maths. No. 819. Springer: Berlin, 1980.Google Scholar
[7]Mailer, M.. Fitted diffeomorphisms on non-simply connected manifolds. Topology 19 (1980), 395–410.CrossRefGoogle Scholar
[8]Manning, A.. There are no new Anosov diffeomorphisms on tori. Amer. J. Math. 96 (1974), 422–429.CrossRefGoogle Scholar
[9]Newhouse, S. E.. On codimension one Anosov diffeomorphisms. Amer. J. Math. 92 (1970), 761–770.CrossRefGoogle Scholar
[10]Shub, M.. Dynamical systems, filtrations and entropy. Bull. Amer. Math. Soc. 80 (1974), 27–41.CrossRefGoogle Scholar
[11]Shub, M. & Sullivan, D.. Homology theory and dynamical systems. Topology 14 (1975), 109–132.CrossRefGoogle Scholar
[12]Sinai, Ja. G.. Construction of Markov partitionings. Func. Anal. Appl. 2 (1968), 39–52.Google Scholar
[13]Smyth, C. J.. On the product of the conjugates outside the unit circle of an algebraic integer. Bull. Lond. Math. Soc. 3 (1971), 169–175.CrossRefGoogle Scholar