2016, 10: 33-79. doi: 10.3934/jmd.2016.10.33

Invariant distributions for homogeneous flows and affine transformations

1. 

UMR CNRS 8524, UFR de Mathématiques, Université de Lille 1, F59655 Villeneuve d’Asq CEDEX

2. 

Department of Mathematics, University of Maryland, College Park, MD 20742-4015, United States

3. 

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802

Received  May 2013 Revised  December 2015 Published  March 2016

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.
Citation: Livio Flaminio, Giovanni Forni, Federico Rodriguez Hertz. Invariant distributions for homogeneous flows and affine transformations. Journal of Modern Dynamics, 2016, 10: 33-79. doi: 10.3934/jmd.2016.10.33
References:
[1]

A. Avila, B. Fayad and A. Kocsard, On manifolds supporting distributionally uniquely ergodic diffeomorphisms,, J. Differential Geom., 99 (2015), 191. Google Scholar

[2]

L. Auslander and L. W. Green, $G$-induced flows,, Amer. J. Math., 88 (1966), 43. doi: 10.2307/2373046. Google Scholar

[3]

A. Avila and A. Kocsard, Cohomological equations and invariant distributions for minimal circle diffeomorphisms,, Duke Math. J., 158 (2011), 501. doi: 10.1215/00127094-1345662. Google Scholar

[4]

_________, Private communication,, in preparation, (2013). Google Scholar

[5]

L. Auslander, An exposition of the structure of solvmanifolds. I. Algebraic theory,, Bull. Amer. Math. Soc., 79 (1973), 227. doi: 10.1090/S0002-9904-1973-13134-9. Google Scholar

[6]

_________, An exposition of the structure of solvmanifolds. II. $G$-induced flows,, Bull. Amer. Math. Soc., 79 (1973), 262. doi: 10.1090/S0002-9904-1973-13139-8. Google Scholar

[7]

W. Chen and M. Y. Chi, Hypoelliptic vector fields and almost periodic motions on the torus $T^ n$,, Commun. Partial Differential Equations, 25 (2000), 337. doi: 10.1080/03605300008821516. Google Scholar

[8]

P. Collet, H. Epstein and G. Gallavotti, Perturbations of geodesic flows on surfaces of constant negative curvature and their mixing properties,, Comm. Math. Phys., 95 (1984), 61. doi: 10.1007/BF01215756. Google Scholar

[9]

S. G. Dani, Spectrum of an affine transformation,, Duke Math. J., 44 (1977), 129. doi: 10.1215/S0012-7094-77-04407-6. Google Scholar

[10]

S. G. Dani, A simple proof of Borel's density theorem,, Math. Z., 174 (1980), 81. doi: 10.1007/BF01215084. Google Scholar

[11]

S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation,, J. Reine Angew. Math., 359 (1985), 55. doi: 10.1515/crll.1985.359.55. Google Scholar

[12]

_________, Bounded orbits of flows on homogeneous spaces,, Comment. Math. Helv., 61 (1986), 636. doi: 10.1007/BF02621936. Google Scholar

[13]

R. de la Llave, J. M. Marco and R. Moriyón, Canonical perturbation theory of Anosov systems and regularity results for Livšic cohomology equation,, Ann. Math. (2), 123 (1986), 537. doi: 10.2307/1971334. Google Scholar

[14]

D. Dolgopyat, Livsič theory for compact group extensions of hyperbolic systems,, Mosc. Math. J., 5 (2005), 55. Google Scholar

[15]

B. Fayad, Rank one and mixing differentiable flows,, Invent. Math., 160 (2005), 305. doi: 10.1007/s00222-004-0408-x. Google Scholar

[16]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows,, Duke Math. J., 119 (2003), 465. doi: 10.1215/S0012-7094-03-11932-8. Google Scholar

[17]

_________, On the cohomological equation for nilflows,, Journal of Modern Dynamics, 1 (2007), 37. Google Scholar

[18]

G. Forni, Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus,, Ann. of Math. (2), 146 (1997), 295. doi: 10.2307/2952464. Google Scholar

[19]

_________, On the Greenfield-Wallach and Katok conjectures in dimension three,, in Geometric and Probabilistic Structures in Dynamics, (2008), 197. doi: 10.1090/conm/469/09167. Google Scholar

[20]

L. Flaminio and M. Paternain, Linearization of cohomology-free vector fields,, Discrete Contin. Dyn. Syst., 29 (2011), 1031. doi: 10.3934/dcds.2011.29.1031. Google Scholar

[21]

V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds,, Topology, 19 (1980), 301. doi: 10.1016/0040-9383(80)90015-4. Google Scholar

[22]

V. V. Gorbatsevich, Splittings of Lie groups and their application to the study of homogeneous spaces,, Math. USSR, 15 (1980), 441. doi: 10.1070/IM1980v015n03ABEH001257. Google Scholar

[23]

S. J. Greenfield and N. R. Wallach, Globally hypoelliptic vector fields,, Topology, 12 (1973), 247. doi: 10.1016/0040-9383(73)90011-6. Google Scholar

[24]

S. Hurder, Problems on rigidity of group actions and cocycles,, Ergodic Theory Dynam. Systems, 5 (1985), 473. doi: 10.1017/S0143385700003084. Google Scholar

[25]

N. Jacobson, Lie Algebras,, Republication of the 1962 original, (1962). Google Scholar

[26]

A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory,, in collaboration with E. A. Robinson, (1999), 107. doi: 10.1090/pspum/069/1858535. Google Scholar

[27]

__________, Combinatorial Constructions in Ergodic Theory and Dynamics,, University Lecture Series, (2003). doi: 10.1090/ulect/030. Google Scholar

[28]

A. Katok and A. Kononenko, Cocycles' stability for partially hyperbolic systems,, Math. Res. Lett., 3 (1996), 191. doi: 10.4310/MRL.1996.v3.n2.a6. Google Scholar

[29]

D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces,, in Sina\u\i's Moscow Seminar on Dynamical Systems, (1996), 141. Google Scholar

[30]

A. Kocsard, Cohomologically rigid vector fields: The Ktwo-formatok conjecture in dimension 3,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1165. doi: 10.1016/j.anihpc.2008.07.005. Google Scholar

[31]

D. Kleinbock, N. Shah and A. Starkov, Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory,, in Handbook of Dynamical Systems, (2002), 813. doi: 10.1016/S1874-575X(02)80013-3. Google Scholar

[32]

A. N. Livšic, Some homology properties of U-systems,, Mat. Zametki, 10 (1971), 555. Google Scholar

[33]

R. Mañé, Contributions to the stability conjecture,, Topology, 17 (1978), 383. doi: 10.1016/0040-9383(78)90005-8. Google Scholar

[34]

S. Matsumoto, The parameter rigid flows on 3-manifolds,, in Foliations, (2009), 135. doi: 10.1090/conm/498/09746. Google Scholar

[35]

S. Marmi, P. Moussa and J.-C. Yoccoz, The cohomological equation for Roth type interval exchange maps,, J. Amer. Math. Soc., 18 (2005), 823. doi: 10.1090/S0894-0347-05-00490-X. Google Scholar

[36]

D. W. Morris, Ratner's Theorems on Unipotent Flows,, Chicago Lectures in Mathematics, (2005). Google Scholar

[37]

A. L. Onishchik and E. B. Vinberg, eds., Lie Groups and Lie Algebras. III. Structure of Lie Groups and Lie Algebras,, A translation of Current Problems in Mathematics. Fundamental Directions, (1990). doi: 10.1007/978-3-662-03066-0. Google Scholar

[38]

W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds,, Amer. J. Math., 91 (1969), 757. doi: 10.2307/2373350. Google Scholar

[39]

M. S. Raghunathan, Discrete Subgroups of Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete, (1972). Google Scholar

[40]

F. Rodriguez Hertz and J. Rodriguez Hertz, Cohomology free systems and the first Betti number,, Continuous and Discrete Dynam. Systems, 15 (2006), 193. doi: 10.3934/dcds.2006.15.193. Google Scholar

[41]

F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle,, Invent. Math., 172 (2008), 353. doi: 10.1007/s00222-007-0100-z. Google Scholar

[42]

A. N. Starkov, On a criterion for the ergodicity of $G$-induced flows,, Uspekhi Mat. Nauk, 42 (1987), 197. Google Scholar

[43]

________, Dynamical Systems on Homogeneous Spaces,, Transl. Math. Monogr., (2000). Google Scholar

[44]

W. A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms,, Ergodic Theory and Dynam. Systems, 6 (1986), 449. doi: 10.1017/S0143385700003606. Google Scholar

[45]

A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms,, Astérisque, 358 (2013), 75. Google Scholar

show all references

References:
[1]

A. Avila, B. Fayad and A. Kocsard, On manifolds supporting distributionally uniquely ergodic diffeomorphisms,, J. Differential Geom., 99 (2015), 191. Google Scholar

[2]

L. Auslander and L. W. Green, $G$-induced flows,, Amer. J. Math., 88 (1966), 43. doi: 10.2307/2373046. Google Scholar

[3]

A. Avila and A. Kocsard, Cohomological equations and invariant distributions for minimal circle diffeomorphisms,, Duke Math. J., 158 (2011), 501. doi: 10.1215/00127094-1345662. Google Scholar

[4]

_________, Private communication,, in preparation, (2013). Google Scholar

[5]

L. Auslander, An exposition of the structure of solvmanifolds. I. Algebraic theory,, Bull. Amer. Math. Soc., 79 (1973), 227. doi: 10.1090/S0002-9904-1973-13134-9. Google Scholar

[6]

_________, An exposition of the structure of solvmanifolds. II. $G$-induced flows,, Bull. Amer. Math. Soc., 79 (1973), 262. doi: 10.1090/S0002-9904-1973-13139-8. Google Scholar

[7]

W. Chen and M. Y. Chi, Hypoelliptic vector fields and almost periodic motions on the torus $T^ n$,, Commun. Partial Differential Equations, 25 (2000), 337. doi: 10.1080/03605300008821516. Google Scholar

[8]

P. Collet, H. Epstein and G. Gallavotti, Perturbations of geodesic flows on surfaces of constant negative curvature and their mixing properties,, Comm. Math. Phys., 95 (1984), 61. doi: 10.1007/BF01215756. Google Scholar

[9]

S. G. Dani, Spectrum of an affine transformation,, Duke Math. J., 44 (1977), 129. doi: 10.1215/S0012-7094-77-04407-6. Google Scholar

[10]

S. G. Dani, A simple proof of Borel's density theorem,, Math. Z., 174 (1980), 81. doi: 10.1007/BF01215084. Google Scholar

[11]

S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation,, J. Reine Angew. Math., 359 (1985), 55. doi: 10.1515/crll.1985.359.55. Google Scholar

[12]

_________, Bounded orbits of flows on homogeneous spaces,, Comment. Math. Helv., 61 (1986), 636. doi: 10.1007/BF02621936. Google Scholar

[13]

R. de la Llave, J. M. Marco and R. Moriyón, Canonical perturbation theory of Anosov systems and regularity results for Livšic cohomology equation,, Ann. Math. (2), 123 (1986), 537. doi: 10.2307/1971334. Google Scholar

[14]

D. Dolgopyat, Livsič theory for compact group extensions of hyperbolic systems,, Mosc. Math. J., 5 (2005), 55. Google Scholar

[15]

B. Fayad, Rank one and mixing differentiable flows,, Invent. Math., 160 (2005), 305. doi: 10.1007/s00222-004-0408-x. Google Scholar

[16]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows,, Duke Math. J., 119 (2003), 465. doi: 10.1215/S0012-7094-03-11932-8. Google Scholar

[17]

_________, On the cohomological equation for nilflows,, Journal of Modern Dynamics, 1 (2007), 37. Google Scholar

[18]

G. Forni, Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus,, Ann. of Math. (2), 146 (1997), 295. doi: 10.2307/2952464. Google Scholar

[19]

_________, On the Greenfield-Wallach and Katok conjectures in dimension three,, in Geometric and Probabilistic Structures in Dynamics, (2008), 197. doi: 10.1090/conm/469/09167. Google Scholar

[20]

L. Flaminio and M. Paternain, Linearization of cohomology-free vector fields,, Discrete Contin. Dyn. Syst., 29 (2011), 1031. doi: 10.3934/dcds.2011.29.1031. Google Scholar

[21]

V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds,, Topology, 19 (1980), 301. doi: 10.1016/0040-9383(80)90015-4. Google Scholar

[22]

V. V. Gorbatsevich, Splittings of Lie groups and their application to the study of homogeneous spaces,, Math. USSR, 15 (1980), 441. doi: 10.1070/IM1980v015n03ABEH001257. Google Scholar

[23]

S. J. Greenfield and N. R. Wallach, Globally hypoelliptic vector fields,, Topology, 12 (1973), 247. doi: 10.1016/0040-9383(73)90011-6. Google Scholar

[24]

S. Hurder, Problems on rigidity of group actions and cocycles,, Ergodic Theory Dynam. Systems, 5 (1985), 473. doi: 10.1017/S0143385700003084. Google Scholar

[25]

N. Jacobson, Lie Algebras,, Republication of the 1962 original, (1962). Google Scholar

[26]

A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory,, in collaboration with E. A. Robinson, (1999), 107. doi: 10.1090/pspum/069/1858535. Google Scholar

[27]

__________, Combinatorial Constructions in Ergodic Theory and Dynamics,, University Lecture Series, (2003). doi: 10.1090/ulect/030. Google Scholar

[28]

A. Katok and A. Kononenko, Cocycles' stability for partially hyperbolic systems,, Math. Res. Lett., 3 (1996), 191. doi: 10.4310/MRL.1996.v3.n2.a6. Google Scholar

[29]

D. Y. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces,, in Sina\u\i's Moscow Seminar on Dynamical Systems, (1996), 141. Google Scholar

[30]

A. Kocsard, Cohomologically rigid vector fields: The Ktwo-formatok conjecture in dimension 3,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1165. doi: 10.1016/j.anihpc.2008.07.005. Google Scholar

[31]

D. Kleinbock, N. Shah and A. Starkov, Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory,, in Handbook of Dynamical Systems, (2002), 813. doi: 10.1016/S1874-575X(02)80013-3. Google Scholar

[32]

A. N. Livšic, Some homology properties of U-systems,, Mat. Zametki, 10 (1971), 555. Google Scholar

[33]

R. Mañé, Contributions to the stability conjecture,, Topology, 17 (1978), 383. doi: 10.1016/0040-9383(78)90005-8. Google Scholar

[34]

S. Matsumoto, The parameter rigid flows on 3-manifolds,, in Foliations, (2009), 135. doi: 10.1090/conm/498/09746. Google Scholar

[35]

S. Marmi, P. Moussa and J.-C. Yoccoz, The cohomological equation for Roth type interval exchange maps,, J. Amer. Math. Soc., 18 (2005), 823. doi: 10.1090/S0894-0347-05-00490-X. Google Scholar

[36]

D. W. Morris, Ratner's Theorems on Unipotent Flows,, Chicago Lectures in Mathematics, (2005). Google Scholar

[37]

A. L. Onishchik and E. B. Vinberg, eds., Lie Groups and Lie Algebras. III. Structure of Lie Groups and Lie Algebras,, A translation of Current Problems in Mathematics. Fundamental Directions, (1990). doi: 10.1007/978-3-662-03066-0. Google Scholar

[38]

W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds,, Amer. J. Math., 91 (1969), 757. doi: 10.2307/2373350. Google Scholar

[39]

M. S. Raghunathan, Discrete Subgroups of Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete, (1972). Google Scholar

[40]

F. Rodriguez Hertz and J. Rodriguez Hertz, Cohomology free systems and the first Betti number,, Continuous and Discrete Dynam. Systems, 15 (2006), 193. doi: 10.3934/dcds.2006.15.193. Google Scholar

[41]

F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle,, Invent. Math., 172 (2008), 353. doi: 10.1007/s00222-007-0100-z. Google Scholar

[42]

A. N. Starkov, On a criterion for the ergodicity of $G$-induced flows,, Uspekhi Mat. Nauk, 42 (1987), 197. Google Scholar

[43]

________, Dynamical Systems on Homogeneous Spaces,, Transl. Math. Monogr., (2000). Google Scholar

[44]

W. A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms,, Ergodic Theory and Dynam. Systems, 6 (1986), 449. doi: 10.1017/S0143385700003606. Google Scholar

[45]

A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms,, Astérisque, 358 (2013), 75. Google Scholar

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