Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T12:33:41.778Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant measures for higher-rank hyperbolic abelian actions

Published online by Cambridge University Press:  19 September 2008

A. Katok  and
R. J. Spatzier
A. Katok
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
R. J. Spatzier
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48103, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate invariant ergodic measures for certain partially hyperbolic and Anosov actions of ℝk, ℤk and We show that they are either Haar measures or that every element of the action has zero metric entropy.

Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 4 , August 1996 , pp. 751 - 778
Copyright
Copyright © Cambridge University Press 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Anosov, D. V.. Geodesic flows on closed manifolds with negative curvature. Proc. Steklov Inst. Maths. 90 (1967).Google Scholar
[2]Berend, D.. Multi-invariant sets on tori. Trans. Amer. Math. Soc. 280 (1983), 509–533.CrossRefGoogle Scholar
[3]Berend, D.. Minimal sets on tori. Ergod. Th. & Dynam. Sys. 4 (1984), 499–507.Google Scholar
[4]Feldman, J.. A generalization of a result of Lyons about measures in (0, 1). Israel J. Math. 81 (1993), 281–287.Google Scholar
[5]Furstenberg, H.. Disjointness in ergodic theory, minimal sets, and a problem in diophantine analysis. Math. Syst. Theory 1 (1967), 1–49.CrossRefGoogle Scholar
[6]Host, B.. Nombres normaux, entropie, translations. Preprint, L.M.D. Marseille.Google Scholar
[7]Huyi, H.. Some ergodic properties of commuting diffeomorphisms. Ergod. Th. & Dynam. Sys. 13 (1993), 73–100.Google Scholar
[8]Johnson, A.. Measures on the circle invariant under multiplication by a nonlacunary subsemigroup of the integers. Israel J. Math. 77 (1992), 211–240.Google Scholar
[9]Katok, A.. Hyperbolicity and rigidity of actions of multi-parameter abelian groups. Workshop on Lie Groups, Ergodic Theory and Geometry. MSRI, Berkeley, 1992, pp. 36–40.Google Scholar
[10]Katok, A. and Katok, S.. Higher cohomology for abelian groups of toral automorphisms. Ergod. Th. & Dynam. Sys. 15 (1995), 569–592.CrossRefGoogle Scholar
[11]Katok, A. and Lewis, J.. Local rigidity for certain groups of toral automorphism. Israel J. Math. 75 (1991), 203–241.Google Scholar
[12]Katok, A. and Schmidt, K.. Cohomology of expanding ℤd-actions by automorphisms of compact groups. Pac. J. Math. 170 (1995), 105–142.Google Scholar
[13]Katok, A. and Spatzier, R. J.. First cohomology of Anosov actions of higher rank Abelian groups and applications to rigidity. Publ. Math. IHES 79 (1994), 131–156.Google Scholar
[14]Katok, A. and Spatzier, R. J.. Subelliptic estimates of polynomial differential operators and applications to cocycle rigidity. Math. Res. Lett. 1 (1994), 193–202.Google Scholar
[15]Katok, A. and Spatzier, R. J.. Differential rigidity of Anosov actions of higher-rank abelian groups and algebraic lattice actions. Preprint, 1996.Google Scholar
[16]Koblitz, N.. p-adic Numbers, p-adic Analysis, and Zeta Functions. Springer, New York, 1984.CrossRefGoogle Scholar
[17]Koppel, N.. Commuting diffeomorphisms. A.M.S. Proc. Symp. Pure Math. 14 (1970), 165–184.Google Scholar
[18]Lang, S.. Algebra. Springer, New York, 1986.Google Scholar
[19]Lang, S.. Algebraic Number Theory. Springer, New York, 1970.Google Scholar
[20]Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. Ann. Math. 122 (1985), 509–539.Google Scholar
[21]Lyons, R.. On measures simultaneously 2- and 3-invariant. Israel J. Math. 61 (1988), 219–224.Google Scholar
[22]Margulis, G. A.. Discrete Subgroups of Lie Groups. Springer, Berlin, 1991.CrossRefGoogle Scholar
[23]Ratner, M.. On Raghunathan's measure conjecture. Ann. Math. 134 (1991), 545–607.Google Scholar
[24]Rees, M.. Some ℝ2-Anosov flows. Preprint, 1982.Google Scholar
[25]Rohklin, V. A.. On the fundamental ideas of measure theory. A.M.S. Transl. (I) 10 (1962), 1–52.Google Scholar
[26]Rudolph, D.. ×2 and ×3 invariant measures and entropy. Ergod. Th. & Dynam. Sys. 10 (1990), 395–406.Google Scholar
[27]Sacksteder, R.. Semigroups of expanding maps. TAMS 221 (1976), 281–288.Google Scholar
[28]Satayev, E. A.. Invariant measures for polynomial semigroups of endomorphisms of the circle. Usp. Math. Nauk 30 (1975), 203–204 (in Russian).Google Scholar