# $C^1$ stability conjecture on non-compact manifolds

In the study of dynamical systems, the link between structural stability and Smale's Axiom A has been explored by many authors. One of the important outcomes of this endeavor was the proof of $C^1$ stability conjecture, which says that a $C^1$-stable system obeys Axiom A. (If I'm not mistaken, the conjecture is a reduction of an earlier conjecture of Palis and Smale.)

A proof for discrete-time dynamical systems was given by Mane [1], and its counterpart for continuous-time dynamical systems (flows) can be found in [2]. Both proofs assume that the state space $M$ is a compact manifold.

What is known for the case of non-compact $M$? Does structural stability still imply some kind of hyperbolicity and density of periodic points (in the non-wandering set)?

Note: My (naive) guess is that the conjecture is probably false for a general non-compact $M$. However, I also expect that a (weakened) version of the conjecture might be salvageable, especially if we make some sort of a boundedness assumption on the orbits of the systems in question.

[1] Mañé, Ricardo. "A proof of the $C^1$ stability conjecture." Publications Mathématiques de l'IHÉS, 66.1 (1987): 161-210.

[2] Wen, Lan. "On the $C^1$ stability conjecture for flows." Journal of Differential Equations 129.2, (1996): 334-357.