Rigidity of measures—The high entropy case and non-commuting foliations
Tóm tắt
We consider invariant measures for partially hyperbolic, semisimple, higher rank actions on homogeneous spaces defined by products of real andp-adic Lie groups. In this paper we generalize our earlier work to establish measure rigidity in the high entropy case in that setting. We avoid any additional ergodicity-type assumptions but rely on, and extend the theory of conditional measures.
Tài liệu tham khảo
D. Berend,Multi-invariant sets on tori, Transactions of the American Mathematical Society280 (1983), 509–532.
D. Berend,Multi-invariant sets on compact abelian groups, Transactions of the American Mathematical Society286 (1984), 505–535.
A. Borel and G. Prasad,Values of isotropic quadratic forms at S-integral points, Compositio Mathematica83 (1992), 347–372.
N. Bourbaki,Lie groups and Lie algebras, Chapters 1–3, inElements of Mathematics, Springer-Verlag, Berlin, 1998. Translated from the French, reprint of the 1989 English translation.
S. G. Dani and J. Smillie,Uniform distribution of horocycle orbits for Fuchsian groups, Duke Mathematical Journal51 (1984), 185–194.
M. Einsiedler and A. Katok,Invariant measures on G/Γ for split simple Lie-groups G, Communications on Pure and Applied Mathematics56 (2003), 1184–1221.
M. Einsiedler, A. Katok, and E. Lindenstrauss,Invariant measures and the set of exceptions to Littlewood's conjecture, Annals of Mathematics (2), to appear.
M. Einsiedler and E. Lindenstrauss,Rigidity properties of ℤ d on tori and solenoids, Electronic Research Announcements of the American Mathematical Society9 (2003), 99–110.
M. Einsiedler and E. Lindenstrauss,Joining of semisimple actions on locally homogeneous spaces, preprint.
J. Feldman,A generalization of a result of R. Lyons about measures on [0, 1), Israel Journal of Mathematics81 (1993), 281–287.
H. Furstenberg,Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Mathematical Systems Theory1 (1967), 1–49.
H. Furstenberg,The unique ergodicity of the horocycle flow, inRecent Advances in Topological Dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Mathematics318, Springer, Berlin, 1973, pp. 95–115.
B. Host,Nombres normaux, entropie, translations, Israel Journal of Mathematics91 (1995), 419–428.
T. Hungerford,Algebra, Holt, Rinehart and Winston, New York, 1974.
A. S. Johnson,Measures on the circle invariant under multiplication by a non-lacunary subsemigroup of the integers, Israel Journal of Mathematics77 (1992), 211–240.
B. Kalinin and A. Katok,Invariant measures for actions of higher rank abelian groups, inSmooth Ergodic Theory and its Applications (Seattle, WA, 1999), American Mathematical Society, Providence, RI, 2001, pp. 593–637.
B. Kalinin and R. Spatzier,Measurable rigidity for higher rank abelian actions, Ergodic Theory and Dynamical Systems25 (2005), 175–200.
A. Katok and R. Spatzier,Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory and Dynamical Systems16 (1996), 751–778.
A. Katok and R. Spatzier,Corrections to: “Invariant measures for higher-rank hyperbolic abelian actions” [Ergodic Theory Dynam. Systems 16 (1996), no. 4, 751–778], Ergodic Theory and Dynamical Systems18 (1998), 503–507.
F. Ledrappier and L.-S. Young,The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Annals of Mathematics (2)122 (1985), 540–574.
E. Lindenstrauss,Invariant measures and arithmetic quantum unique ergodicity, Annals of Mathematics (2), to appear.
E. Lindenstrauss and B. Weiss,On sets invariant under the action of the diagonal group, Ergodic Theory and Dynamical Systems21 (2001), 1481–1500.
R. Lyons,On measures simultaneously 2- and 3-invariant, Israel Journal of Mathematics61 (1988), 219–224.
G. Margulis,Discrete subgroups and ergodic theory, inNumber Theory, Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press, Boston, 1989, pp. 377–398.
G. Margulis, and G. Tomanov,Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Inventiones Mathematicae116 (1994), 347–392.
W. Parry,Squaring and cubing the circle—Rudolph's theorem, inErgodic Theory of ℤ d Actions (Warwick, 1993–1994), London Mathematical Society Lecture Note Series, Vol. 228, Cambridge University Press, Cambridge, 1996, pp. 177–183.
M. Ratner,On Raghunathan's measure conjecture, Annals of Mathematics (2)134 (1991), 545–607.
M. Ratner,Raghunathan's topological conjecture and distributions of unipotent flows, Duke Mathematical Journal63 (1991), 235–280.
M. Ratner,Raghunathan's conjectures for SL(2,R), Israel Journal of Mathematics80 (1992), 1–31.
M. Ratner,Interactions between ergodic theory, Lie groups, and number theory, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, pp. 157–182.
M. Ratner,Raghunathan's conjectures for Cartesian products of real and p-adic Lie groups, Duke Mathematical Journal77 (1995), 275–382.
M. Rees,Some ℕ 2 flows, Unpublished manuscript, 1982.
D. J. Rudolph,×2 and ×3 invariant measures and entropy, Ergodic Theory and Dynamical Systems10 (1990), 395–406.