Geodesic
Rigidity properties of ℤ^{𝕕}-actions on tori and solenoids
Einsiedler, Manfred1 ; Lindenstrauss, Elon2, 3
1 Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195
2 Department of Mathematics, Stanford University, Stanford, CA 94305
3 Courant Institute of Mathematical Sciences, 251 Mercer St., New York, NY 10012
Electronic research announcements of the American Mathematical Society, Tome 09 (2003), pp. 99-110.

Voir la notice de l'article provenant de la source American Mathematical Society

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.
DOI : 10.1090/S1079-6762-03-00117-3

Einsiedler, Manfred 1 ; Lindenstrauss, Elon 2, 3

1 Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195
2 Department of Mathematics, Stanford University, Stanford, CA 94305
3 Courant Institute of Mathematical Sciences, 251 Mercer St., New York, NY 10012 
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Einsiedler, Manfred; Lindenstrauss, Elon. Rigidity properties of ℤ^{𝕕}-actions on tori and solenoids. Electronic research announcements of the American Mathematical Society, Tome 09 (2003), pp. 99-110. doi : 10.1090/S1079-6762-03-00117-3. http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-03-00117-3/

[1] Berend, Daniel Multi-invariant sets on tori Trans. Amer. Math. Soc. 1983 509 532

[2] Berend, Daniel Multi-invariant sets on compact abelian groups Trans. Amer. Math. Soc. 1984 505 535

[3] Feldman, J. A generalization of a result of R. Lyons about measures on [0,1) Israel J. Math. 1993 281 287

[4] Furstenberg, Harry Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation Math. Systems Theory 1967 1 49

[5] Host, Bernard Nombres normaux, entropie, translations Israel J. Math. 1995 419 428

[6] Hu, Hu Yi Some ergodic properties of commuting diffeomorphisms Ergodic Theory Dynam. Systems 1993 73 100

[7] Johnson, Aimee S. A. Measures on the circle invariant under multiplication by a nonlacunary subsemigroup of the integers Israel J. Math. 1992 211 240

[8] Kalinin, Boris, Katok, Anatole Invariant measures for actions of higher rank abelian groups 2001 593 637

[9] Katok, Anatole, Katok, Svetlana, Schmidt, Klaus Rigidity of measurable structure for ℤ^{𝕕}-actions by automorphisms of a torus Comment. Math. Helv. 2002 718 745

[10] Katok, A., Spatzier, R. J. Invariant measures for higher-rank hyperbolic abelian actions Ergodic Theory Dynam. Systems 1996 751 778

[11] Katok, A., Spatzier, R. J. Corrections to: “Invariant measures for higher-rank hyperbolic abelian actions” [Ergodic Theory Dynam. Systems 16 (1996), no. 4, 751–778 Ergodic Theory Dynam. Systems 1998 503 507

[12] Ledrappier, F., Young, L.-S. The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula Ann. of Math. (2) 1985 509 539

[13] Ledrappier, F., Young, L.-S. The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula Ann. of Math. (2) 1985 509 539

[14] Lindenstrauss, Elon đť‘ť-adic foliation and equidistribution Israel J. Math. 2001 29 42

[15] Lindenstrauss, Elon, Meiri, David, Peres, Yuval Entropy of convolutions on the circle Ann. of Math. (2) 1999 871 904

[16] Lyons, Russell On measures simultaneously 2- and 3-invariant Israel J. Math. 1988 219 224

[17] Oxtoby, J. C., Ulam, S. M. On the existence of a measure invariant under a transformation Ann. of Math. (2) 1939 560 566

[18] Rudolph, Daniel J. Ă—2 and Ă—3 invariant measures and entropy Ergodic Theory Dynam. Systems 1990 395 406

[19] Schmidt, Klaus Dynamical systems of algebraic origin 1995

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