eBook ISBN:  9781470400682 
Product Code:  MEMO/103/491.E 
List Price:  $40.00 
MAA Member Price:  $36.00 
AMS Member Price:  $24.00 
eBook ISBN:  9781470400682 
Product Code:  MEMO/103/491.E 
List Price:  $40.00 
MAA Member Price:  $36.00 
AMS Member Price:  $24.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 103; 1993; 138 ppMSC: Primary 58; Secondary 57
Let \(F:M\rightarrow M\) denote a selfdiffeomorphism 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.
ReadershipResearch mathematicians.

Table of Contents

Chapters

1. Introduction

2. Some linear constructions

3. Proofs of Propositions 2.10 and 2.14

4. Some smooth constructions

5. The foliation hypothesis

6. Smooth triangulation near $\Lambda $

7. Smooth ball structures near $\Lambda $

8. Triangulating image balls

9. The thickening theorem

10. Results in P.L. topology

11. Proof of the thickening theorem

12. The limit theorem

13. Construction of Markov cells

14. Removing the foliation hypothesis

15. Selected problems


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Let \(F:M\rightarrow M\) denote a selfdiffeomorphism 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.

Chapters

1. Introduction

2. Some linear constructions

3. Proofs of Propositions 2.10 and 2.14

4. Some smooth constructions

5. The foliation hypothesis

6. Smooth triangulation near $\Lambda $

7. Smooth ball structures near $\Lambda $

8. Triangulating image balls

9. The thickening theorem

10. Results in P.L. topology

11. Proof of the thickening theorem

12. The limit theorem

13. Construction of Markov cells

14. Removing the foliation hypothesis

15. Selected problems