Hyperbolicity test and structural stability

The aim of the paper is the substantiation of a constructive method for verification of hyperbolicity and structural stability of discrete dynamical systems. The main tool to do so is a symbolic image which is a directed graph constructed by a finite covering of the projective bundle. Hyperbolicity is tested by the calculation of the Morse spectrum (the limit set of Lyapunov exponents of pseudo trajectories) which can be found for a given accuracy by the construction of a symbolic image (3). If the Morse spectrum does not contain 0, then the chain recurrent set is hyperbolic and the system is Omega-stable. Thus, the symbolic image gives us an opportunity to verify these properties. A diffeomorphism f is shown to be structurally stable if and only if the Morse spectrum does not contain 0 and for the complementary differential there is no connection CR+ -> CR- on the projective bundle. These conditions are verified by an algorithm based on the symbolic image of the complementary differential.