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A discrete dynamical system generated by a homeomorphism of a compact manifold is considered. A sequence of periodic -trajectories converges in the mean as if, for any continuous function , the mean values on the period converge as . It is shown that converges in the mean if and only if there exists an invariant measure such that converges to . If a sequence converges in the mean and converges uniformly to a trajectory , then the trajectory is recurrent and its closure is a minimal strictly ergodic set. [ABSTRACT FROM AUTHOR]

We consider a method of applied symbolic dynamics which may be used to obtain wide spectrum of characteristics of complex dynamical systems: approximation of invariant sets, estimation of topological entropy and approximation of invariant measures. The method has received wide acceptance in studying complex dynamical systems. The main idea of the method is to describe the system behaviour approximately by means of an oriented graph (called symbolic image). Such a graph is a representation of symbolic dynamical system which is more appropriately known as topological Markov chain. There exists a correspondence between trajectories of a given system and paths on the graph, which allows us to design algorithms for estimation of topological entropy, approximate invariant measures of a given system by using stationary flows on the graph and calculate corresponding metric entropies. These values characterize complex behaviour of dynamical systems such as the existence of trajectories with large periods and chaotic regimes. The results of experiments are given for systems with chaotic dynamics. [ABSTRACT FROM AUTHOR]

A discrete dynamical system generated by a diffeomorphism f on a compact manifold is considered. The Morse spectrum is the limit set of Lyapunov exponents of periodic pseudotrajectories. It is proved that the Morse spectrum coincides with the set of averagings of the function ϕ(x, e) = ln| Df( x) e| over the invariant measures of the mapping induced by the differential Df on the projective bundle. [ABSTRACT FROM AUTHOR]