# Markov Cell Structures near a Hyperbolic Set

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*Tom Farrell; Lowell Jones*

Let \(F:M\rightarrow M\) denote a self-diffeomorphism of the smooth manifold \(M\) and let \(\Lambda \subset M\) denote a hyperbolic set for \(F\). Roughly speaking, a Markov cell structure for \(F:M\rightarrow M\) near \(\Lambda\) is a finite cell structure \(C\) for a neighborhood of \(\Lambda\) in \(M\) such that, for each cell \(e \in C\), the image under \(F\) of the unstable factor of \(e\) is equal to the union of the unstable factors of a subset of \(C\), and the image of the stable factor of \(e\) under \(F^{-1}\) is equal to the union of the stable factors of a subset of \(C\). The main result of this work is that for some positive integer \(q\), the diffeomorphism \(F^q:M\rightarrow M\) has a Markov cell structure near \(\Lambda\). A list of open problems related to Markov cell structures and hyperbolic sets can be found in the final section of the book.

#### Table of Contents

# Table of Contents

## Markov Cell Structures near a Hyperbolic Set

- Contents v6 free
- 1. Introduction 18 free
- 2. Some Linear Constructions 1320 free
- 3. Proofs of Propositions 2.10 and 2.14 1825
- 4. Some Smooth Constructions 3643
- 5. The Foliation Hypothesis 4047
- 6. Smooth Triangulation Near Λ 4451
- 7. Smooth Ball Structures Near Λ 4956
- 8. Triangulating Image Balls 6168
- 9. The Thickening Theorem 6976
- 10. Results in P.L. Topology 7279
- 11. Proof of the Thickening Theorem 8087
- 12. The Limit Theorem 9198
- 13. Construction of Markov Cells 104111
- 14. Removing the Foliation Hypothesis 109116
- 15. Selected Problems 126133
- References 137144