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?