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Smooth Markov partitions and toral automorphisms

Published online by Cambridge University Press:  19 September 2008

Elise Cawley
Elise Cawley
Affiliation:
Mathematical Sciences Research Institute, Berkeley, CA 94720, USA
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Abstract

We show that the only hyperbolic toral automorphisms f for which there exist Markov partitions with piecewise smooth boundary are those for which a power fk is linearly covered by a direct product of automorphisms of the 2-torus. Only a finite number of shapes occur in a certain natural set of cross-sections of the partition boundary. The behavior of the stratified structure of a piecewise smooth boundary under the mapping forces these shapes to be self-similar. This, together with expanding properties of the mapping, means that a piecewise smooth partition is in fact piecewise linear. Orbits of affine disks in the boundary are used to construct a basis of 2-dimensional invariant toral subgroups, and then the product decomposition of a covering follows easily.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 11 , Issue 4 , December 1991 , pp. 633 - 651
Copyright
Copyright © Cambridge University Press 1991

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References

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