Hostname: page-component-6bf8c574d5-mggfc Total loading time: 0 Render date: 2025-02-20T19:47:43.618Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[A]Alexandroff, P.. Dimensionstheorie. Math. Ann. 106 (1932), 161–238.CrossRefGoogle Scholar
[B]Bowen, R.. Markov partitions are not smooth. Proc. Amer. Math. Soc. 71 (1978), 130–132.CrossRefGoogle Scholar
[FJ]Farrell, T. & Jones, L.. Markov cell structures near a hyperbolic set. Preprint.CrossRefGoogle Scholar
[HW]Hurewicz, W. and Wallman, H.. Dimension Theory. Princeton University Press, Princeton, NJ (1948).Google Scholar
[K]King, H.. Topological invariance of intersection homology without sheaves. Topology and its Applications 20 (1985), 149–160.CrossRefGoogle Scholar
[Ko]Kodama, Y.. On a problem of Alexandroff concerning the dimension of product spaces I. J. Math. Soc. Japan. 10 (1958), 380–404.Google Scholar
[RS]Rourke, C. P. & Sanderson, B. J.. Introduction to Piecewise Linear Topology. Springer-Verlag (1970).Google Scholar
[S]Sinai, Ya.. Construction of Markov Partitions. Functional Anal. Appl. 2 (1968), 70–80.CrossRefGoogle Scholar
[Sb]Siebenmann, L. C.. Deformation of homeomorphisms on stratified sets. Commun. Math. Helv. 47 (1972), 123–163.CrossRefGoogle Scholar