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

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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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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.
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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