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.

Readership

Research mathematicians.

Table of Contents

• 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