Abstract
Let X be a metrizable space and let ϕ:ℝ × X → X be a continuous flow on X. For any given {φt}–invariant Borel probability measure, this paper presents a {ϕ t }–invariant Borel subset of X satisfying the requirements of the classical ergodic theorem for the continuous flow (X, {ϕ t }). The set is more restrictive than the ones in the literature, but it might be more useful and convenient, particularly for non–uniformly hyperbolic systems and skew–product flows.
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References
Nemytskii, V., Stepanov, V.: Qualitative Theory of Di.erential Equations, Princeton University Press, New York, 1960
Walters, P.: An Introduction to Ergodic Theory, GTM 79, Springer-Verlag, New York, 1982
Liao, S. T.: An ergodic property theorem for a di.erential system. Scientia Sinica, 16(1), 1–24 (1973)
Dai, X.: On the continuity of Liao qualitative functions of di.erential systems and applications. Commun. Contemp. Math., to appear
Liao, S. T.: On characteristic exponents construction of a new Borel set for the Multiplicative Ergodic Theorem for vector fields. Acta Scientiarum Naturalium Univ. Pekinensis, 29(3), 277–302 (1993)
Liao, S. T.: Notes on a study of vector bundle dynamical systems (II)-part 1. Appl. Math. Mechanics, 17(9), 805–818 (1996)
Parthasarthy, K. R.: Introduction to Probability and Measure, Macmilian Press LTD, New York, 1977
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Dai is partially supported by NSF Grant (C19901029). Zhou is supported in part by NSF Grant (10171116)
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Dai, X.P., Zhou, Z.L. A Generalization of a Theorem of Liao. Acta Math Sinica 22, 207–210 (2006). https://doi.org/10.1007/s10114-005-0617-2
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DOI: https://doi.org/10.1007/s10114-005-0617-2