The spectrum of the averaging of a function over pseudotrajectories of a dynamical system

The paper is concerned with the discrete dynamical system generated by a homeomorphism $f$ on a compact manifold $M$ and with a continuous function $\varphi$. The averaging of $\varphi$ over a periodic $\varepsilon$-trajectory is the arithmetic mean of the values of $\varphi$ on the period. The limit set as $\varepsilon \to 0$ of the averagings over periodic $\varepsilon$-trajectories is called the spectrum of the averaging. The spectrum is shown to consist of closed intervals, where each interval is generated by a component of the chain recurrent set and can be obtained by averaging the function $\varphi$ over all invariant measures concentrated on this component.
Bibliography: 18 titles.