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

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SUMMARY

The discussion centers on proving that Morse-Smale (MS) diffeomorphisms are dense and open in the space of diffeomorphisms on the circle, Diff(S1). Key definitions include finite chain recurrence sets and Kupka-Smale conditions, which involve transversality and hyperbolic periodic points. The proof strategy suggested involves constructing a sequence of diffeomorphisms f1, f2, and f3, each with specific properties related to periodic points and hyperbolicity, as outlined in De Strien's book. The goal is to develop a proof that is both precise and avoids overly complex methods.

PREREQUISITES
  • Understanding of Morse-Smale diffeomorphisms
  • Familiarity with the concept of finite chain recurrence sets
  • Knowledge of Kupka-Smale conditions in dynamical systems
  • Basic principles of hyperbolic dynamics
NEXT STEPS
  • Study the construction of Morse-Smale diffeomorphisms in detail
  • Learn about the properties of non-wandering sets in dynamical systems
  • Investigate the implications of hyperbolicity in periodic points
  • Review De Strien's book for specific examples and proof techniques
USEFUL FOR

This discussion is beneficial for mathematicians, particularly those specializing in dynamical systems, as well as graduate students exploring the properties of diffeomorphisms and their applications in topology and analysis.

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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