Structural stability of a dynamical system near a non-hyperbolic fixed point

摘要

We prove structural stability under perturbations for a class ofdiscrete-time dynamical systems near a non-hyperbolic fixed point. Wereformulate the stability problem in terms of the well-posedness of aninfinite-dimensional nonlinear ordinary differential equation in a Banach spaceof carefully weighted sequences. Using this, we prove existence and regularityof flows of the dynamical system which obey mixed initial and final boundaryconditions. The class of dynamical systems we study, and the boundaryconditions we impose, arise in a renormalisation group analysis of the4-dimensional weakly self-avoiding walk and the 4-dimensional n-component$|\varphi|^4$ spin model.