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Dynamical properties of quasihyperbolic toral automorphisms

Published online by Cambridge University Press:  13 August 2009

D. A. Lind
D. A. Lind
Affiliation:
Department of Mathematics, University of Washington, Seattle, Washington 98195
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Abstract

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We study the dynamical properties of ergodic toral autmorphisms that have some eigenvalues of modulus one. For such automorphisms, all sufficiently fine smooth partitions generate measurably, but never topologically, and are never weak Bernoulli. The points of period k become uniformly distributed exponentially fast, and Lipschitz functions mix exponentially fast. Every reasonably smooth compact null set has the property that there is a dense set of periodic points whose entire orbit misses the set, but this is false for general compact null sets. Katznelson's property of almost weak Bernoulli can be strengthened to a certain exponential rate of independence, but breaks down at a critical number. Finally, open sets have return times that decay exponentially fast.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 2 , Issue 1 , March 1982 , pp. 49 - 68
Copyright
Copyright © Cambridge University Press 1982

References

REFERENCES

[1]Adler, R. L. & Weiss, B.. Similarity of Automorphisms of the Torus. Memoirs Amer. Math. Soc. 98 (1970).Google Scholar
[2]Bowen, Rufus. Periodic points and measures for Axiom A diffeomorphisms. Trans. Amer. Math. Soc. 154 (1971), 377–397.Google Scholar
[3]Bowen, Rufus. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Math. no. 470. Springer: Berlin, 1975.CrossRefGoogle Scholar
[4]Bowen, Rufus. Smooth partitions of Anosov diffeomorphisms are weak Bernoulli. Israel J. Math. 21 (1975), 95–100.CrossRefGoogle Scholar
[5]del Junco, A. & Rahe, M.. Finitary codings and weak Bernoulli partitions. Proc. Amer. Math. Soc. 75 (1974), 259–264.CrossRefGoogle Scholar
[6]Gelfond, A. O.. Transcendental and Algebraic Numbers. Dover: New York, 1960.Google Scholar
[7]Katznelson, Yitzhak. An Introduction to Harmonic Analysis. Wiley: New York, 1968.Google Scholar
[8]Katznelson, Yitzhak. Ergodic automorphisms of Tn are Bernoulli shifts. Israel J. Math. 10 (1971), 186–195.CrossRefGoogle Scholar
[9]Lind, D. A.. Ergodic Group Automorphisms and Specification. Lecture Notes in Math. no. 729, pp. 93–104. Springer: Berlin, 1979.Google Scholar
[10]Lind, D. A.. Finitarily splitting skew products. In Ergodic Theory and Dynamical Systems I (Katok, A. B., ed.), pp. 65–80. Birkhäuser: Boston, 1981.CrossRefGoogle Scholar
[11]Marcus, Brian. A note on periodic points of toral automorphisms. Monatsh. Math. 89 (1980), 121–129.CrossRefGoogle Scholar
[12]Ornstein, Donald. Ergodic Theory, Randomness, and Dynamical Systems. Yale Math. Monographs 5. Yale University Press: New Haven, 1974.Google Scholar
[13]Peters, Justin. A discrete analogue of a theorem of Katznelson. Israel J. Math. 37 (1980), 251–255.CrossRefGoogle Scholar
[14]Rohlin, V. A.. On the fundamental ideas of measure theory. Amer. Math. Soc. Translations, Ser. 1, 10 (1962), 1–54.Google Scholar
[15]Rudolph, Daniel J.. A characterization of those processes finitarily isomorphic to a Bernoulli shift. In Ergodic Theory and Dynamical Systems I (Katok, A. B., ed.). pp. 1–64. Birkhäuser: Boston, 1981.Google Scholar
[16]Shields, Paul C.. Weak and very weak Bernoulli partitions. Monatsh. Math. 84 (1972), 133–142.CrossRefGoogle Scholar
[17]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
[18]Smorodinsky, M.. A partition on a Bernoulli shift which is not weak Bernoulli. Math. Syst. Th. 5 (1971), 201–203.CrossRefGoogle Scholar
[19]Stolarsky, Kenneth B.. Algebraic Numbers and Diophantine Approximation. Dekker: New York, 1974.Google Scholar