# $C^r$ Stability Conjecture

**Conjecture**Any 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 the conjecture was proved by R Mañé (Publ. IHES 1986). In higher regularity, , 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 topology has been proved by S Hayashi (Ann Math. 1997).

## Bibliography

* indicates original appearance(s) of problem.