$\rho(|DF|)<1$ without global asymptotic stability]{Maps with $\rho(|DF|)<1$ need not have a globally asymptotically stable fixed point

Let $F$ be a $C^1$ map of $\R^n$ with a fixed point, and let $|DF(x)|$ be the Jacobian matrix of $F$ at $x$ with each entry replaced by its absolute value. The condition that the spectral radius of $|DF(x)|$ is less than $1$ at every $x$ makes the fixed point globally asymptotically stable, that is, stable and the limit of every orbit, for instance if $n=1$ or if $F$ is polynomial. Cima, Gasull and Mañosas asked whether the condition does so for every $C^1$ map with a fixed point. Gasull asked the same question for smooth maps of the plane. The answer to both questions is negative. We construct a $C^\infty$ map and a real-analytic map of the plane for which the spectral radius of $|DF|$ is bounded by a constant less than $1$. Each map has a fixed point and an orbit of period three, which does not converge to the fixed point. At every point, one of the entries off the diagonal of the $2\times2$ Jacobian matrix of the $C^\infty$ map vanishes, so the spectral radius of $|DF|$ is the larger absolute value of the diagonal entries. Products with the map $w\mapsto w/2$ of $\R^{n-2}$ give such maps in every dimension $n\ge3$.