# 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), 191-213. [2] L. Auslander and L. W. Green, $G$-induced flows, Amer. J. Math., 88 (1966), 43-60. doi: 10.2307/2373046. [3] A. Avila and A. Kocsard, Cohomological equations and invariant distributions for minimal circle diffeomorphisms, Duke Math. J., 158 (2011), 501-536. 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), 227-261. 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), 262-285. 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), 337-354. 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), 61-112. doi: 10.1007/BF01215756. [9] S. G. Dani, Spectrum of an affine transformation, Duke Math. J., 44 (1977), 129-155. doi: 10.1215/S0012-7094-77-04407-6. [10] S. G. Dani, A simple proof of Borel's density theorem, Math. Z., 174 (1980), 81-94. doi: 10.1007/BF01215084. [11] S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., 359 (1985), 55-89. doi: 10.1515/crll.1985.359.55. [12] _________, Bounded orbits of flows on homogeneous spaces, Comment. Math. Helv., 61 (1986), 636-660. 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), 537-611. doi: 10.2307/1971334. [14] D. Dolgopyat, Livsič theory for compact group extensions of hyperbolic systems, Mosc. Math. J., 5 (2005), 55-67. [15] B. Fayad, Rank one and mixing differentiable flows, Invent. Math., 160 (2005), 305-340. 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), 465-526. doi: 10.1215/S0012-7094-03-11932-8. [17] _________, On the cohomological equation for nilflows, Journal of Modern Dynamics, 1 (2007), 37-60. [18] G. Forni, Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus, Ann. of Math. (2), 146 (1997), 295-344. doi: 10.2307/2952464. [19] _________, On the Greenfield-Wallach and Katok conjectures in dimension three, in Geometric and Probabilistic Structures in Dynamics, Contemp. Math., 469, Amer. Math. Soc., Providence, RI, 2008, 197-213. doi: 10.1090/conm/469/09167. [20] L. Flaminio and M. Paternain, Linearization of cohomology-free vector fields, Discrete Contin. Dyn. Syst., 29 (2011), 1031-1039. 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), 301-312. 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, Izv., 15 (1980), 441-467. doi: 10.1070/IM1980v015n03ABEH001257. [23] S. J. Greenfield and N. R. Wallach, Globally hypoelliptic vector fields, Topology, 12 (1973), 247-254. 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), 473-484. doi: 10.1017/S0143385700003084. [25] N. Jacobson, Lie Algebras, Republication of the 1962 original, Dover Publications, Inc., New York, 1979. [26] A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory, in collaboration with E. A. Robinson, Jr., in Smooth Ergodic Theory and its Applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001, 107-173. doi: 10.1090/pspum/069/1858535. [27] __________, Combinatorial Constructions in Ergodic Theory and Dynamics, University Lecture Series, 30, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/ulect/030. [28] A. Katok and A. Kononenko, Cocycles' stability for partially hyperbolic systems, Math. Res. Lett., 3 (1996), 191-210. 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, Amer. Math. Soc. Transl. Ser. 2, 171, Amer. Math. Soc., Providence, RI, 1996, 141-172. [30] A. Kocsard, Cohomologically rigid vector fields: The Ktwo-formatok conjecture in dimension 3, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1165-1182. 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, Vol. 1A, North-Holland, Amsterdam, 2002, 813-930. doi: 10.1016/S1874-575X(02)80013-3. [32] A. N. Livšic, Some homology properties of U-systems, Mat. Zametki, 10 (1971), 555-564. [33] R. Mañé, Contributions to the stability conjecture, Topology, 17 (1978), 383-396. doi: 10.1016/0040-9383(78)90005-8. [34] S. Matsumoto, The parameter rigid flows on 3-manifolds, in Foliations, Geometry, and Topology, Contemp. Math., 498, Amer. Math. Soc., Providence, RI, 2009, 135-139. 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), 823-872. doi: 10.1090/S0894-0347-05-00490-X. [36] D. W. Morris, Ratner's Theorems on Unipotent Flows, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 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, Vol. 41, (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. (VINITI), Moscow, 1990; Encyclopaedia of Mathematical Sciences, 41, Springer-Verlag, Berlin, 1994. 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), 757-771. doi: 10.2307/2373350. [39] M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. [40] F. Rodriguez Hertz and J. Rodriguez Hertz, Cohomology free systems and the first Betti number, Continuous and Discrete Dynam. Systems, 15 (2006), 193-196. 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), 353-381. 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), 197-198. [43] ________, Dynamical Systems on Homogeneous Spaces, Transl. Math. Monogr., 190, Amer. Math. Soc., Providence, 2000. [44] W. A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms, Ergodic Theory and Dynam. Systems, 6 (1986), 449-473. doi: 10.1017/S0143385700003606. [45] A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, Astérisque, 358 (2013), 75-165.

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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), 191-213. [2] L. Auslander and L. W. Green, $G$-induced flows, Amer. J. Math., 88 (1966), 43-60. doi: 10.2307/2373046. [3] A. Avila and A. Kocsard, Cohomological equations and invariant distributions for minimal circle diffeomorphisms, Duke Math. J., 158 (2011), 501-536. 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), 227-261. 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), 262-285. 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), 337-354. 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), 61-112. doi: 10.1007/BF01215756. [9] S. G. Dani, Spectrum of an affine transformation, Duke Math. J., 44 (1977), 129-155. doi: 10.1215/S0012-7094-77-04407-6. [10] S. G. Dani, A simple proof of Borel's density theorem, Math. Z., 174 (1980), 81-94. doi: 10.1007/BF01215084. [11] S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. Reine Angew. Math., 359 (1985), 55-89. doi: 10.1515/crll.1985.359.55. [12] _________, Bounded orbits of flows on homogeneous spaces, Comment. Math. Helv., 61 (1986), 636-660. 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), 537-611. doi: 10.2307/1971334. [14] D. Dolgopyat, Livsič theory for compact group extensions of hyperbolic systems, Mosc. Math. J., 5 (2005), 55-67. [15] B. Fayad, Rank one and mixing differentiable flows, Invent. Math., 160 (2005), 305-340. 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), 465-526. doi: 10.1215/S0012-7094-03-11932-8. [17] _________, On the cohomological equation for nilflows, Journal of Modern Dynamics, 1 (2007), 37-60. [18] G. Forni, Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus, Ann. of Math. (2), 146 (1997), 295-344. doi: 10.2307/2952464. [19] _________, On the Greenfield-Wallach and Katok conjectures in dimension three, in Geometric and Probabilistic Structures in Dynamics, Contemp. Math., 469, Amer. Math. Soc., Providence, RI, 2008, 197-213. doi: 10.1090/conm/469/09167. [20] L. Flaminio and M. Paternain, Linearization of cohomology-free vector fields, Discrete Contin. Dyn. Syst., 29 (2011), 1031-1039. 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), 301-312. 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, Izv., 15 (1980), 441-467. doi: 10.1070/IM1980v015n03ABEH001257. [23] S. J. Greenfield and N. R. Wallach, Globally hypoelliptic vector fields, Topology, 12 (1973), 247-254. 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), 473-484. doi: 10.1017/S0143385700003084. [25] N. Jacobson, Lie Algebras, Republication of the 1962 original, Dover Publications, Inc., New York, 1979. [26] A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory, in collaboration with E. A. Robinson, Jr., in Smooth Ergodic Theory and its Applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001, 107-173. doi: 10.1090/pspum/069/1858535. [27] __________, Combinatorial Constructions in Ergodic Theory and Dynamics, University Lecture Series, 30, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/ulect/030. [28] A. Katok and A. Kononenko, Cocycles' stability for partially hyperbolic systems, Math. Res. Lett., 3 (1996), 191-210. 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, Amer. Math. Soc. Transl. Ser. 2, 171, Amer. Math. Soc., Providence, RI, 1996, 141-172. [30] A. Kocsard, Cohomologically rigid vector fields: The Ktwo-formatok conjecture in dimension 3, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1165-1182. 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, Vol. 1A, North-Holland, Amsterdam, 2002, 813-930. doi: 10.1016/S1874-575X(02)80013-3. [32] A. N. Livšic, Some homology properties of U-systems, Mat. Zametki, 10 (1971), 555-564. [33] R. Mañé, Contributions to the stability conjecture, Topology, 17 (1978), 383-396. doi: 10.1016/0040-9383(78)90005-8. [34] S. Matsumoto, The parameter rigid flows on 3-manifolds, in Foliations, Geometry, and Topology, Contemp. Math., 498, Amer. Math. Soc., Providence, RI, 2009, 135-139. 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), 823-872. doi: 10.1090/S0894-0347-05-00490-X. [36] D. W. Morris, Ratner's Theorems on Unipotent Flows, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 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, Vol. 41, (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. (VINITI), Moscow, 1990; Encyclopaedia of Mathematical Sciences, 41, Springer-Verlag, Berlin, 1994. 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), 757-771. doi: 10.2307/2373350. [39] M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. [40] F. Rodriguez Hertz and J. Rodriguez Hertz, Cohomology free systems and the first Betti number, Continuous and Discrete Dynam. Systems, 15 (2006), 193-196. 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), 353-381. 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), 197-198. [43] ________, Dynamical Systems on Homogeneous Spaces, Transl. Math. Monogr., 190, Amer. Math. Soc., Providence, 2000. [44] W. A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms, Ergodic Theory and Dynam. Systems, 6 (1986), 449-473. doi: 10.1017/S0143385700003606. [45] A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, Astérisque, 358 (2013), 75-165.
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