Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I am asking for a reference that contains a proof of Theorem 4, which is on page 315 of the following text:

Hirsch, Morris W., and Stephen Smale. Differential equations, dynamical systems, and linear algebra. Vol. 60. Academic press, 1974.

Let $W$ be an open set in a vector space and $\mathcal{V}(W)$ be the set of all $C^1$ vector fields on $W$. Let $D^n = \{ x \in \mathbb{R}^n : \lvert x \rvert \leq 1 \}$. Consider in $\mathcal{V}(D^n)$ the set $\mathrm{grad}(D^n)$ of gradient vector fields that point inward on $D^n$.

Theorem 4 The set of structurally stable systems contained in $\mathrm{grad}(D^n)$ is open and dense in $\mathrm{grad}(D^n)$.

share|improve this question
add comment

1 Answer

up vote 1 down vote accepted

This came out of J. Palis' 1967 Thesis:

J. Palis "On Morse-Smale dynamical systems" Topology 8, 1969, 385--405.

But that dealt with dimension $\leq 3$. The result you mention seems to first appear as a corollary in

J. Palis and S. Smale "Structural stability theorems" in Global analysis proceedings Symp. Pure Math., 14 AMS, 1970, 223--231.

share|improve this answer
    
Using your information to start my search, I found an even earlier reference in: Smale, Stephen. "On gradient dynamical systems." The Annals of Mathematics 74.1 (1961): 199-206. in the form of Theorem A –  James Rohal Feb 8 '13 at 2:19
    
I thought I didn't do that bad from home (without library access to the papers). Thanks for completing the reference hunt :) –  Rodrigo A. Pérez Feb 8 '13 at 2:52
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.