Is Morse-Smale Dense and Open in Diff(S1)?

  • Thread starter arpharazon
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In summary, the conversation is about proving that the Morse-Smale diffeomorphism is dense and open in Diff(S1), using an adapted proof without using a hammer. The suggested approach is to take a point p in the non-wandering set of f and find three diffeomorphisms (f1, f2, f3) that are close to f and have p as a periodic point, with f2 and f3 being hyperbolic and all periodic points of f3 being hyperbolic. The question is whether there is any logic in this approach and how to prove the original statement. Any suggestions and incomplete answers are welcome.
  • #1
arpharazon
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Hi everybody,

Taking as a general definition of Morse-Smale (MS) diffeo:

- finite chain recurrence set
- Kupka-smale (ie transversalit +hyperbolic periodic points)

How would you proove that MS is dense and open in Diff(S1)?

The goal is to have an adapted proof, not use a hammer.

There is de strien book who asks to:

Take p in non-wandering set of f.

- find f1 close to f with p in Per(f1)
- find f2 with p in Per(f2) and hyperbolic
- find f3 with p in Per(f3) and all of its periodic points are hyperbolic

Can you see any logic in this? How would you prove the original statement?Thanks for your help! :)
 
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  • #2
Any suggestions are most welcome, even if they are incomplete answers...
 

FAQ: Is Morse-Smale Dense and Open in Diff(S1)?

What is Morse-Smale dense in Diff?

Morse-Smale dense in Diff is a mathematical concept that describes a dense subset of diffeomorphisms, which are smooth transformations between differentiable manifolds. It is used in the study of dynamical systems and the behavior of chaotic systems.

How is Morse-Smale dense in Diff related to Morse theory?

Morse theory is a mathematical tool that helps analyze the topology of a smooth manifold by studying the critical points of a smooth function on that manifold. Morse-Smale dense in Diff is closely related to Morse theory as it also uses the critical points of a smooth function to describe the dynamics of a diffeomorphism.

What is the significance of Morse-Smale dense in Diff in the study of dynamical systems?

Morse-Smale dense in Diff provides a way to understand the behavior of chaotic systems and how they evolve over time. It can help identify the stable and unstable sets of a dynamical system, which are important in predicting the long-term behavior of the system.

How is Morse-Smale dense in Diff used in applications?

Morse-Smale dense in Diff has applications in a variety of fields, including physics, biology, and engineering. It can be used to study the dynamics of fluid flows, the behavior of biological systems, and the stability of mechanical systems.

What are some current research topics related to Morse-Smale dense in Diff?

Some current research topics include generalizing Morse-Smale dense in Diff to higher dimensions, studying the stability of periodic orbits in chaotic systems, and using Morse-Smale theory to analyze the dynamics of networks and neural systems.

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