TY - JOUR

T1 - Measurable rigidity of the cohomological equation for linear cocycles over hyperbolic systems

AU - Butler, Clark

N1 - Funding Information:
∗ This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant # DGE-1144082. Received April 8, 2016 and in revised form March 1, 2017
Publisher Copyright:
© 2018, Hebrew University of Jerusalem.

PY - 2018/8/1

Y1 - 2018/8/1

N2 - We show that any measurable solution of the cohomological equation for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a Hölder solution. More generally, we show that every measurable invariant conformal structure for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a continuous invariant conformal structure. We also use the main theorem to show that a linear cocycle is conformal if none of its iterates preserve a measurable family of proper subspaces of Rd. We use this to characterize closed negatively curved Riemannian manifolds of constant negative curvature by irreducibility of the action of the geodesic flow on the unstable bundle.

AB - We show that any measurable solution of the cohomological equation for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a Hölder solution. More generally, we show that every measurable invariant conformal structure for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a continuous invariant conformal structure. We also use the main theorem to show that a linear cocycle is conformal if none of its iterates preserve a measurable family of proper subspaces of Rd. We use this to characterize closed negatively curved Riemannian manifolds of constant negative curvature by irreducibility of the action of the geodesic flow on the unstable bundle.

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U2 - 10.1007/s11856-018-1733-x

DO - 10.1007/s11856-018-1733-x

M3 - Article

AN - SCOPUS:85050291569

SN - 0021-2172

VL - 227

SP - 27

EP - 61

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -