TY - JOUR
T1 - Invariant distributions for homogeneous flows and affine transformations
AU - Flaminio, Livio
AU - Forni, Giovanni
AU - Hertz, Federico Rodriguez
N1 - 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 -