Lorenz attractors do not have the pseudo-orbit tracing property (Q801579)

The paper investigates a pseudo-orbit tracing property (POTP) for a one- parameter flow on the geometric Lorenz attractor which is a modeled strange attractor of a flow on \(R^ 3\) defined by J. Guckenheimer. The main result is the following: (i) The geometric Lorenz attractors do not have the finite POTP, except a special case such that the return map f on [0,1] satisfies \(f(0)=0\) and \(f(1)=1\). (ii) The set of all vector fields whose flows satisfy the strong POTP is not dense in the space of all \(C^ 2\)-vector fields on a compact 3-manifold. This gives an answer of flow case for A. Morimoto's problem whether or not the set of all diffeomorphisms with POTP is dense in the space of all diffeomorphisms on a compact manifold.