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

Let F be a C¹ map of 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 C¹ 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^∞ 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 2×2 Jacobian matrix of the C^∞ map vanishes, so the spectral radius of $|DF|$ is the larger absolute value of the diagonal entries. Products with the map w↦ w/2 of ⁿ⁻² give such maps in every dimension n≥3.