Duke Mathematical Journal

On topological and measurable dynamics of unipotent frame flows for hyperbolic manifolds

François Maucourant and Barbara Schapira

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We study the dynamics of unipotent flows on frame bundles of hyperbolic manifolds of infinite volume. We prove that they are topologically transitive and that the natural invariant measure, the so-called Burger–Roblin measure, is ergodic, as soon as the geodesic flow admits a finite measure of maximal entropy, and this entropy is strictly greater than the codimension of the unipotent flow inside the maximal unipotent flow. The latter result generalizes a theorem of Mohammadi and Oh.

Article information

Duke Math. J., Volume 168, Number 4 (2019), 697-747.

Received: 2 May 2017
Revised: 19 October 2018
First available in Project Euclid: 5 February 2019

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Zentralblatt MATH identifier

Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Secondary: 37C45: Dimension theory of dynamical systems 37A40: Nonsingular (and infinite-measure preserving) transformations 28D20: Entropy and other invariants 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]

unipotent flows ergodicity Burger–Roblin measure mixing frame flows Bowen–Margulis–Sullivan measure Marstrand’s theorem


Maucourant, François; Schapira, Barbara. On topological and measurable dynamics of unipotent frame flows for hyperbolic manifolds. Duke Math. J. 168 (2019), no. 4, 697--747. doi:10.1215/00127094-2018-0050. https://projecteuclid.org/euclid.dmj/1549392546

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