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We relate the existence of many infinite geodesics on Alexandrov spaces to a statement about the average growth of volumes of balls.We deduce that the geodesic flow exists and preserves the Liouville measure in several important cases. The analytic tools we develop have close ties to integral geometry.
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Vitali Kapovitch, Alexander Lytchak, Anton Petrunin, Metric-measure boundary and geodesic flow on Alexandrov spaces. J. Eur. Math. Soc. 23 (2021), no. 1, pp. 29–62DOI 10.4171/JEMS/1006