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Generating special Markov partitions for hyperbolic toral automorphisms using fractals

Published online by Cambridge University Press:  19 September 2008

Tim Bedford
Tim Bedford
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16, Mill Lane, Cambridge, England
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Abstract

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We show that given some natural conditions on a 3 × 3 hyperbolic matrix of integers A(det A = 1) there exists a Markov partition for the induced map A(x + ℤ3) = A(x)+ℤ3 on T3 whose transition matrix is (A−1)t. For expanding endomorphisms of T2 we construct a Markov partition so that there is a semiconjugacy from a full (one-sided) shift.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 6 , Issue 3 , September 1986 , pp. 325 - 333
Copyright
Copyright © Cambridge University Press 1986

References

REFERENCES

[1]Adler, R. L. and Marcus, B.. Topological equivalence of dynamical systems. Mem. Amer. Math. Soc. 219 (1979).Google Scholar
[2]Bedford, T. J.. Ph.D. Thesis, Warwick University (1984).Google Scholar
[3]Bedford, T. J.. Dimension and dynamics for fractal recurrent sets. Preprint, King's College Research Centre, Cambridge (1985).Google Scholar
[4]Bowen, R.. Equilibrium States and the Ergodic Theory of Ansov Diffeomorphisms. Lect. Notes in Maths. Vol 470. Springer-Verlag: Berlin, 1975.CrossRefGoogle Scholar
[5]Bowen, R.. Markov partitions are not smooth. Proc. Amer. Math. Soc. 71 (1978) 130–132.CrossRefGoogle Scholar
[6]Dekking, F. M.. Recurrent sets. Advances in Math. 44 (1982) 78–104.CrossRefGoogle Scholar
[7]Dekking, F. M.. Replicating superfigures and endomorphisms of free groups. J. Combin. Thry. Ser. A 32 (1982) 315–320.CrossRefGoogle Scholar
[8]Mañé, R.. Orbits of paths under hyperbolic toral automorphisms. Proc. Amer. Math. Soc. 73, 121–125.CrossRefGoogle Scholar
[9]Manning, A. K.. Lecture notes for M.Sc course on Dynamical Systems (1983).Google Scholar
[10]Urbanski, M.. On capacity of continuum with non-dense orbit under a hyperbolic toral automorphism.xPreprint 289, Inst. of Maths., Polish Academy of Sciences, November (1983).Google Scholar