Markov Cell Structures Near a Hyperbolic Set

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.