# American Institute of Mathematical Sciences

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. [2] L. Auslander and L. W. Green, $G$-induced flows,, Amer. J. Math., 88 (1966), 43. doi: 10.2307/2373046. [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. [4] _________, Private communication,, in preparation, (2013). [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. [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. [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. [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. [9] S. G. Dani, Spectrum of an affine transformation,, Duke Math. J., 44 (1977), 129. doi: 10.1215/S0012-7094-77-04407-6. [10] S. G. Dani, A simple proof of Borel's density theorem,, Math. Z., 174 (1980), 81. doi: 10.1007/BF01215084. [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. [12] _________, Bounded orbits of flows on homogeneous spaces,, Comment. Math. Helv., 61 (1986), 636. doi: 10.1007/BF02621936. [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. [14] D. Dolgopyat, Livsič theory for compact group extensions of hyperbolic systems,, Mosc. Math. J., 5 (2005), 55. [15] B. Fayad, Rank one and mixing differentiable flows,, Invent. Math., 160 (2005), 305. doi: 10.1007/s00222-004-0408-x. [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. [17] _________, On the cohomological equation for nilflows,, Journal of Modern Dynamics, 1 (2007), 37. [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. [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. [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. [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. [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. [23] S. J. Greenfield and N. R. Wallach, Globally hypoelliptic vector fields,, Topology, 12 (1973), 247. doi: 10.1016/0040-9383(73)90011-6. [24] S. Hurder, Problems on rigidity of group actions and cocycles,, Ergodic Theory Dynam. Systems, 5 (1985), 473. doi: 10.1017/S0143385700003084. [25] N. Jacobson, Lie Algebras,, Republication of the 1962 original, (1962). [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. [27] __________, Combinatorial Constructions in Ergodic Theory and Dynamics,, University Lecture Series, (2003). doi: 10.1090/ulect/030. [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. [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. [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. [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. [32] A. N. Livšic, Some homology properties of U-systems,, Mat. Zametki, 10 (1971), 555. [33] R. Mañé, Contributions to the stability conjecture,, Topology, 17 (1978), 383. doi: 10.1016/0040-9383(78)90005-8. [34] S. Matsumoto, The parameter rigid flows on 3-manifolds,, in Foliations, (2009), 135. doi: 10.1090/conm/498/09746. [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. [36] D. W. Morris, Ratner's Theorems on Unipotent Flows,, Chicago Lectures in Mathematics, (2005). [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. [38] W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds,, Amer. J. Math., 91 (1969), 757. doi: 10.2307/2373350. [39] M. S. Raghunathan, Discrete Subgroups of Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete, (1972). [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. [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. [42] A. N. Starkov, On a criterion for the ergodicity of $G$-induced flows,, Uspekhi Mat. Nauk, 42 (1987), 197. [43] ________, Dynamical Systems on Homogeneous Spaces,, Transl. Math. Monogr., (2000). [44] W. A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms,, Ergodic Theory and Dynam. Systems, 6 (1986), 449. doi: 10.1017/S0143385700003606. [45] A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms,, Astérisque, 358 (2013), 75.

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##### References:
 [1] A. Avila, B. Fayad and A. Kocsard, On manifolds supporting distributionally uniquely ergodic diffeomorphisms,, J. Differential Geom., 99 (2015), 191. [2] L. Auslander and L. W. Green, $G$-induced flows,, Amer. J. Math., 88 (1966), 43. doi: 10.2307/2373046. [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. [4] _________, Private communication,, in preparation, (2013). [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. [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. [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. [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. [9] S. G. Dani, Spectrum of an affine transformation,, Duke Math. J., 44 (1977), 129. doi: 10.1215/S0012-7094-77-04407-6. [10] S. G. Dani, A simple proof of Borel's density theorem,, Math. Z., 174 (1980), 81. doi: 10.1007/BF01215084. [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. [12] _________, Bounded orbits of flows on homogeneous spaces,, Comment. Math. Helv., 61 (1986), 636. doi: 10.1007/BF02621936. [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. [14] D. Dolgopyat, Livsič theory for compact group extensions of hyperbolic systems,, Mosc. Math. J., 5 (2005), 55. [15] B. Fayad, Rank one and mixing differentiable flows,, Invent. Math., 160 (2005), 305. doi: 10.1007/s00222-004-0408-x. [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. [17] _________, On the cohomological equation for nilflows,, Journal of Modern Dynamics, 1 (2007), 37. [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. [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. [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. [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. [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. [23] S. J. Greenfield and N. R. Wallach, Globally hypoelliptic vector fields,, Topology, 12 (1973), 247. doi: 10.1016/0040-9383(73)90011-6. [24] S. Hurder, Problems on rigidity of group actions and cocycles,, Ergodic Theory Dynam. Systems, 5 (1985), 473. doi: 10.1017/S0143385700003084. [25] N. Jacobson, Lie Algebras,, Republication of the 1962 original, (1962). [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. [27] __________, Combinatorial Constructions in Ergodic Theory and Dynamics,, University Lecture Series, (2003). doi: 10.1090/ulect/030. [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. [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. [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. [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. [32] A. N. Livšic, Some homology properties of U-systems,, Mat. Zametki, 10 (1971), 555. [33] R. Mañé, Contributions to the stability conjecture,, Topology, 17 (1978), 383. doi: 10.1016/0040-9383(78)90005-8. [34] S. Matsumoto, The parameter rigid flows on 3-manifolds,, in Foliations, (2009), 135. doi: 10.1090/conm/498/09746. [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. [36] D. W. Morris, Ratner's Theorems on Unipotent Flows,, Chicago Lectures in Mathematics, (2005). [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. [38] W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds,, Amer. J. Math., 91 (1969), 757. doi: 10.2307/2373350. [39] M. S. Raghunathan, Discrete Subgroups of Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete, (1972). [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. [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. [42] A. N. Starkov, On a criterion for the ergodicity of $G$-induced flows,, Uspekhi Mat. Nauk, 42 (1987), 197. [43] ________, Dynamical Systems on Homogeneous Spaces,, Transl. Math. Monogr., (2000). [44] W. A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms,, Ergodic Theory and Dynam. Systems, 6 (1986), 449. doi: 10.1017/S0143385700003606. [45] A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms,, Astérisque, 358 (2013), 75.
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