# $C^r$ 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

\begin{conjecture} Any $C^r$ structurally stable diffeomorphism is hyperbolic. \end{conjecture}

See the definitions of: \href[stractural stability]{http://en.wikipedia.org/wiki/Structural_stability} and \href[hyperbolicity]{http://en.wikipedia.org/wiki/Hyperbolic_structure}.

The conjecture is due to J Palis and S Smale (1970's). In the case $r=1$ the conjecture was proved by \href[R Mañé (Publ. IHES 1986)]{http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1987__66__161_0}. 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

% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

* indicates original appearance(s) of problem.