The relation between a flow and its discretization

It is proved that the h-time map of a hyperbolic flow and its h -discretization are uniformly topologically conjugate for each small positive h . Introduction. Let $(£, x) be the flow generated by the equation x = Ax + g(x), (1) where A 6 £ ( R m ) , A has no eigenvalues on the imaginary axis, g £ C(R,R), g(0) = 0, sup\g\ 0, then the mapping (2) has local stable and unstable manifolds TV£, W£ for the fixed point 0, respectively. Recently the author of this paper [1] has shown that the manifolds W£, W£ tend to Wy W u as h • 0, h > 0, where W% W are local stable, unstable manifolds of (1) for the fixed point 0, respectively. The purpose of this paper is to show that the mapping 3>(/i, -) and G(/i, •) are uniformly topologically conjugate for each small positive h, i.e. the following theorem holds: A M S S u b j e c t C l a s s i f i c a t i o n (1991): Primary 58F99. Secondary 34C35. K e y w o r d s : Discretization, Dynamical systems, Hartman-Grobman theorem.