A convex function satisfying the Łojasiewicz inequality but failing the gradient conjecture both at zero and infinity

Abstract

We construct an example of a smooth convex function on the plane with a strict minimum at zero, which is real analytic except at zero, for which Thom's gradient conjecture fails both at zero and infinity. More precisely, the gradient orbits of the function spiral around zero and at infinity. Besides, the function satisfies the Łojasiewicz gradient inequality at zero.