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$C^r$ Stability Conjecture

Importance: Outstanding ✭✭✭✭
Author(s): Palis, J.
Smale, S.
Subject: Analysis
Keywords: diffeomorphisms,
dynamical systems
Recomm. for undergrads: no
Posted by: m n
on: December 20th, 2007
Conjecture   Any $ C^r $ structurally stable diffeomorphism is hyperbolic.

See the definitions of: stractural stability and hyperbolicity.

The conjecture is due to J Palis and S Smale (1970's). In the case $ r=1 $ the conjecture was proved by R Mañé (Publ. IHES 1986). In higher regularity, $ r>1 $, the conjecture is one of the most important and difficult problems in dynamical systems.

There is a similar conjecture for the vector fields or flows, and in the $ C^1 $ topology has been proved by S Hayashi (Ann Math. 1997).

Bibliography



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