Rigidity properties of $\mathbb {Z}^d$-actions on tori and solenoids
Authors:
Manfred Einsiedler and Elon Lindenstrauss
Journal:
Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 99-110
MSC (2000):
Primary 37A35; Secondary 37A45
DOI:
https://doi.org/10.1090/S1079-6762-03-00117-3
Published electronically:
October 14, 2003
MathSciNet review:
2029471
Full-text PDF Free Access
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Abstract: We show that Haar measure is a unique measure on a torus or more generally a solenoid $X$ invariant under a not virtually cyclic totally irreducible $\mathbb Z^d$-action by automorphisms of $X$ such that at least one element of the action acts with positive entropy. We also give a corresponding theorem in the non-irreducible case. These results have applications regarding measurable factors and joinings of these algebraic $\mathbb Z^d$-actions.
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Einsiedler-Lind Manfred Einsiedler and Doug Lind, Algebraic ${\mathbb Z}^ d$-actions of entropy rank one, to appear in Trans. Amer. Math. Soc.
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Einsiedler-Schmidt Manfred Einsiedler and Klaus Schmidt, Irreducibility, homoclinic points and adjoint actions of algebraic $\mathbb {Z}^ d$-actions of rank one, Dynamics and randomness (Santiago, 2000), Nonlinear Phenom. Complex Systems 7 (2002), 95–124.
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Berend-invariant-tori Daniel Berend, Multi-invariant sets on tori, Trans. Amer. Math. Soc. 280 (1983), no. 2, 509–532.
Berend-invariant-groups —, Multi-invariant sets on compact abelian groups, Trans. Amer. Math. Soc. 286 (1984), no. 2, 505–535.
Einsiedler-Katok Manfred Einsiedler and Anatole Katok, Invariant measures on $G \backslash \Gamma$ for split simple Lie groups $G$, Comm. Pure Appl. Math. 56 (2003), no. 8, 1184–1221.
Einsiedler-Lind Manfred Einsiedler and Doug Lind, Algebraic ${\mathbb Z}^ d$-actions of entropy rank one, to appear in Trans. Amer. Math. Soc.
Einsiedler-Lindenstrauss Manfred Einsiedler and Elon Lindenstrauss, Rigidity properties of measure preserving $\mathbb Z^d$-actions on tori and solenoids, in preparation.
Einsiedler-Schmidt Manfred Einsiedler and Klaus Schmidt, Irreducibility, homoclinic points and adjoint actions of algebraic $\mathbb {Z}^ d$-actions of rank one, Dynamics and randomness (Santiago, 2000), Nonlinear Phenom. Complex Systems 7 (2002), 95–124.
Feldman-generalization J. Feldman, A generalization of a result of R. Lyons about measures on $[0,1)$, Israel J. Math. 81 (1993), no. 3, 281–287.
Furstenberg-disjointness-1967 Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49.
Host-normal-numbers Bernard Host, Nombres normaux, entropie, translations, Israel J. Math. 91 (1995), no. 1-3, 419–428.
Hu-commuting-diffeomorphisms Hu Yi Hu, Some ergodic properties of commuting diffeomorphisms, Ergodic Theory Dynam. Systems 13 (1993), no. 1, 73–100.
Johnson-invariant-measures Aimee S. A. Johnson, Measures on the circle invariant under multiplication by a nonlacunary subsemigroup of the integers, Israel J. Math. 77 (1992), no. 1-2, 211–240.
Kalinin-Spatzier B. Kalinin and R. J. Spatzier, Measurable rigidity for higher rank abelian actions, preprint.
Kalinin-Katok-Seattle Boris Kalinin and Anatole Katok, Invariant measures for actions of higher rank abelian groups, Smooth ergodic theory and its applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., Providence, RI, 2001, pp. 593–637.
Kalinin-Katok —, Measurable rigidity and disjointness for $\mathbb Z^ k$ actions by toral automorphisms, Ergodic Theory Dynam. Systems 22 (2002), no. 2, 507–523.
Katok-Katok-Schmidt Anatole Katok, Svetlana Katok, and Klaus Schmidt, Rigidity of measurable structure for ${\mathbb {Z}}^ d$-actions by automorphisms of a torus, Comment. Math. Helv. 77 (2002), no. 4, 718–745.
Katok-Spatzier A. Katok and R. J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory Dynam. Systems 16 (1996), no. 4, 751–778.
Katok-Spatzier-corrections —, Corrections to: “Invariant measures for higher-rank hyperbolic abelian actions” [Ergodic Theory Dynam. Systems 16 (1996), no. 4, 751–778; MR 97d:58116], Ergodic Theory Dynam. Systems 18 (1998), no. 2, 503–507.
Ledrappier-Young-I F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2) 122 (1985), no. 3, 509–539.
Ledrappier-Young-II —, The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. of Math. (2) 122 (1985), no. 3, 540–574.
Lindenstrauss-p-adic Elon Lindenstrauss, $p$-adic foliation and equidistribution, Israel J. Math. 122 (2001), 29–42.
Lindenstrauss-Quantum —, Invariant measures and arithmetic quantum unique ergodicity, preprint.
Lindenstrauss-Meiri-Peres Elon Lindenstrauss, David Meiri, and Yuval Peres, Entropy of convolutions on the circle, Ann. of Math. (2) 149 (1999), no. 3, 871–904.
Lyons-2-and-3 Russell Lyons, On measures simultaneously $2$- and $3$-invariant, Israel J. Math. 61 (1988), no. 2, 219–224.
Parry-2-3 William Parry, Squaring and cubing the circle—Rudolph’s theorem, Ergodic theory of ${\mathbb Z}^ d$ actions (Warwick, 1993–1994), London Math. Soc. Lecture Note Ser., vol. 228, Cambridge Univ. Press, Cambridge, 1996, pp. 177–183.
Rudolph-2-and-3 Daniel J. Rudolph, $\times 2$ and $\times 3$ invariant measures and entropy, Ergodic Theory Dynam. Systems 10 (1990), no. 2, 395–406.
Schmidt-book Klaus Schmidt, Dynamical systems of algebraic origin, Progress in Mathematics, vol. 128, Birkhäuser Verlag, Basel, 1995.
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Additional Information
Manfred Einsiedler
Affiliation:
Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195
MR Author ID:
636562
Email:
einsiedl@math.washington.edu
Elon Lindenstrauss
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305
Address at time of publication:
Courant Institute of Mathematical Sciences, 251 Mercer St., New York, NY 10012
MR Author ID:
605709
Email:
elonbl@member.ams.org
Keywords:
Entropy,
invariant measures,
invariant $\sigma$-algebras,
measurable factors,
joinings,
toral automorphisms,
solenoid automorphism
Received by editor(s):
July 12, 2003
Published electronically:
October 14, 2003
Additional Notes:
E.L. is supported in part by NSF grant DMS-0140497. The two authors gratefully acknowledge the hospitality of Stanford University and the University of Washington, respectively
Communicated by:
Klaus Schmidt
Article copyright:
© Copyright 2003
American Mathematical Society