Spectrum of a Dynamical System and Applied Symbolic Dynamics1

The paper introduces a constructive method for localization of the Morse spectrum of a dynamical system on a vector bundle. The Morse spectrum is a limit set of Lyapunov exponents of periodic pseudo-trajectories. The proposed method does not demand any preliminary information on a system. An induced dynamical system on the projective bundle is associated with a directed graph called the symbolic image. The symbolic image can be considered as a finite discrete approximation of a dynamical system. Valuable information about the system may come from the analysis of a symbolic image. In particular, a neighborhood of the Morse spectrum can be found. A special sequence of symbolic images is considered to obtain a sequence of embedded neighborhoods which converges to the Morse spectrum. The main results of this article were announced in a previous paper Ž‘‘Proceedings of the Fifteenth IMACS World Congress,’’ 1997, Vol. 1, pp. 15 30..