Geodesic
The spectrum of the averaging of a~function over pseudotrajectories of a~dynamical system
Sbornik. Mathematics, Tome 209 (2018) no. 8, pp. 1211-1233

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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.
Keywords: pseudotrajectory, chain recurrent component, symbolic image, invariant measure, flow on graph. 
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     author = {G. S. Osipenko},
     title = {The spectrum of the averaging of a~function over pseudotrajectories of a~dynamical system},
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G. S. Osipenko. The spectrum of the averaging of a~function over pseudotrajectories of a~dynamical system. Sbornik. Mathematics, Tome 209 (2018) no. 8, pp. 1211-1233. http://geodesic.mathdoc.fr/item/SM_2018_209_8_a4/