# How Markus–Yamabe implies Jacobian ?

To make myself precise, I would like to recall some backgrounds.

(Markus-Yamabe, $\mathrm{MY}_n$) Given a $C^1$ map $f:\Bbb R^n \to \Bbb R^n$ with $f(0)=0$ and $Df$ everywhere Hurwitz stable (the real part of all eigenvalues of $Df$ are $<0$), 0 is the global attractor of dynamical system $\dot{x}=f(x)$.

$\mathrm{MY}_2$ is now a theorem (Fessler; Glutsyuk; Gutierrez), and a polynomial counterexample has been found for $\mathrm{MY}_3$ (Cima et al.)

I would also like to distinguish 2 kinds of "real Jacobian conjecture":

(real Jacobian, $\mathrm{RJ}_n$) Given a polynomial map $f:\Bbb R^n \to \Bbb R^n$, the Jacobian $J_f$ being some non-zero constant implies $f$ is a diffeomorphism. I believe a stronger statement can be made by taking $f$ to be $C^1$.

(strong Jacobian, $\mathrm{SJ}_n$) Given a $C^1$ map $f:\Bbb R^n \to \Bbb R^n$, $J_f(x)>0$ implies $f$ is a diffeomorphism.

Notice that Pinchuk has found a polynomial counterexample for $\mathrm{SJ}_2$.

I learnt recently a very vague statement that the Markus-Yamabe conjecture implies the real Jacobian conjecture. I think these stuffs must be well-known among the experts, so could anyone

(1)make this statement rigorous by indicating the relationship between $\mathrm{MY}_n$, $\mathrm{RJ}_n$ and $\mathrm{SJ}_n$?

(2)or even better, show me how the argument goes,

(3)or locate some reference?

Thanks a lot!

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Have you looked at the book `Polynomial Automorphism' by Arno Van den Essen? He certainly mentions several things about M-Y conjecture. – Mohan Aug 22 '12 at 19:18