TY - JOUR
T1 - Invariant distributions for homogeneous flows and affine transformations
AU - Flaminio, Livio
AU - Forni, Giovanni
AU - Hertz, Federico Rodriguez
N1 - Funding Information:
The authors wish to thank A. Katok for several comments on the first version of this paper which led to a significant improvement of the exposition and of the results. L. Flaminio was supported in part by the Labex CEMPI (ANR-11-LABX-07). G. Forni was supported by NSF grant DMS 1201534. F. Rodriguez Hertz was supported by NSF grant DMS 1201326. L. Flaminio would also like to thank the Department Mathematics of the University of Maryland, College Park, for its hospitality during the preparation of this paper.
Publisher Copyright:
© 2016 AIMSCIENCES.
PY - 2016/3/22
Y1 - 2016/3/22
N2 - We prove that every homogeneous flow on a finite-volume homogeneous manifold has countably many independent invariant distributions unless it is conjugate to a linear flow on a torus. We also prove that the same conclusion holds for every affine transformation of a homogenous space which is not conjugate to a toral translation. As a part of the proof, we have that any smooth partially hyperbolic flow on any compact manifold has countably many distinct minimal sets, hence countably many distinct ergodic probability measures. As a consequence, the Katok and Greenfield-Wallach conjectures hold in all of the above cases.
AB - We prove that every homogeneous flow on a finite-volume homogeneous manifold has countably many independent invariant distributions unless it is conjugate to a linear flow on a torus. We also prove that the same conclusion holds for every affine transformation of a homogenous space which is not conjugate to a toral translation. As a part of the proof, we have that any smooth partially hyperbolic flow on any compact manifold has countably many distinct minimal sets, hence countably many distinct ergodic probability measures. As a consequence, the Katok and Greenfield-Wallach conjectures hold in all of the above cases.
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U2 - 10.3934/jmd.2016.10.33
DO - 10.3934/jmd.2016.10.33
M3 - Article
AN - SCOPUS:84961616383
SN - 1930-5311
VL - 10
SP - 33
EP - 79
JO - Journal of Modern Dynamics
JF - Journal of Modern Dynamics
ER -