Summary
Let T be a one-parameter semigroup of measure preserving transformations of a probability space. The theorem of Kac on mean recurrence time of the points x in a measurable subset E under a discrete semigroup is carried over to the case of a flow T with continuous time parameter t≧0. Recurrence time is defined as the infimum of these parameter values t>0 for which the orbit of x has returned to E after having temporarily left the set E. The results are first formulated for a probability space without any topological structure; they are then applied to the case of a continuous flow in a compact metric space.
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Helmberg, G. Über mittlere Rückkehrzeit unter einer ma\treuen Strömung. Z. Wahrscheinlichkeitstheorie verw Gebiete 13, 165–179 (1969). https://doi.org/10.1007/BF00537022
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DOI: https://doi.org/10.1007/BF00537022