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Markov Cell Structures near a Hyperbolic Set
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Memoirs of the American Mathematical Society
1993; 138 pp; softcover
Volume: 103
ISBN-10: 0-8218-2553-4
ISBN-13: 978-0-8218-2553-2
List Price: US$40 Individual Members: US$24
Institutional Members: US\$32
Order Code: MEMO/103/491

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.

Research mathematicians.

• Some linear constructions
• Proofs of propositions 2.10 and 2.14
• Some smooth constructions
• The foliation hypothesis
• Smooth triangulation near $$\Lambda$$
• Smooth ball structures near $$\Lambda$$
• Triangulating image balls
• The thickening theorem
• Results in P. L. topology
• Proof of the thickening theorem
• The limit theorem
• Construction of Markov cells
• Removing the foliation hypothesis
• Selected problems