Duke Mathematical Journal

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

François Maucourant and Barbara Schapira

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1549392546

Digital Object Identifier
doi:10.1215/00127094-2018-0050

Mathematical Reviews number (MathSciNet)
MR3916066

Zentralblatt MATH identifier
07055153

Subjects
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]

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

Citation

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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