Let $U$ be an open subset of $\mathbb{R}^2$. For my purposes, we can assume that $U$ is just a rectangle. I have an infinitely differentiable map $M:U\to U$ that has a unique fixed point $p$ in $U$. Furthermore, the Jacobian of $M$ at $p$ only has eigenvalues with absolute value less than $1$, so I know that $p$ is a globally attracting fixed point in some neighborhood of $p$.


I also know that the Jacobian of $M$ taken at every other point in $U$ also only has eigenvalues with absolute value less than $1$. Does this imply that the $M$-orbit of every point in $U$ must converge to $p$?


EDIT: What happens if we insist that the $U$ must be convex?



Jacobian has small eigenvalues everywhere in an open set. Does this imply globally attracting fixed point in that set?
No consider $U=\{(x,y)\in \mathbb{R}^2| |xy|<1/4\}$, $M(x,y)=(y^2,x^2)$ then $(0,0)$ is the only fixpoint but the orbit of $(1,0)$ does not converge to $(0,0)$.


