Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Isomorphism rigidity of commuting automorphisms

Author(s): Siddhartha Bhattacharya
Journal: Trans. Amer. Math. Soc. 360 (2008), 6319-6329.
MSC (2000): Primary 37A35, 37A15
Posted: July 24, 2008
MathSciNet review: 2434289
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Let , and let $(X,\alpha)$ and $(Y,\beta)$ be two zero-entropy ${\mathbb{Z}}^d$ -actions on compact abelian groups by commuting automorphisms. We show that if all lower rank subactions of $\alpha$ and $\beta$ have completely positive entropy, then any measurable equivariant map from to is an affine map. In particular, two such actions are measurably conjugate if and only if they are algebraically conjugate.

References:

1.: S. Bhattacharya, Zero entropy algebraic ${\mathbf{Z}}^d$ -actions that do not exhibit rigidity, Duke Math. J., vol. 116, 471-476, 2003. MR 1958095 (2004c:37010)
2.: S. Bhattacharya, Higher order mixing and rigidity of algebraic actions on compact abelian groups, Israel J. Math., vol. 137, 211-221, 2003. MR 2013357 (2004g:37030)
3.: S. Bhattacharya and K. Schmidt, Homoclinic points and isomorphism rigidity of algebraic ${\mathbf{Z}}^d$ -actions on zero dimensional compact abelian groups, Israel J. Math., vol. 137, 189-209, 2003. MR 2013356 (2004g:37029)
4.: S. Bhattacharya and T. Ward, Finite entropy characterizes topological rigidity, Ergodic Theory Dynam. Systems, vol. 25, 365-373, 2005. MR 2129101 (2006g:37003)
5.: R. Bieri and J.R.J. Groves, The geometry of the set of characters induced by valuations, J. Reine Angew. Math., vol. 347, 168-195, 1984. MR 733052 (86c:14001)
6.: R. Bieri and R. Strebel, Valuations and finitely presented metabelian groups, Proc. London Math. Soc. (3), vol. 41, 439-464, 1980. MR 591649 (81j:20080)
7.: D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Springer-Verlag, New York, 1995. MR 1322960 (97a:13001)
8.: M. Einsiedler, Isomorphism and measure rigidity for algebraic actions on zero-dimensional groups, Monatsh. Math., vol. 144, 39-69, 2005. MR 2109928 (2005i:37008)
9.: M. Einsiedler and E. Lindenstrauss, Rigidity properties of ${\mathbb{Z}}^d$ -actions on tori and solenoids, Electron. Res. Announc. Amer. Math. Soc., vol. 9, 99-110, 2003. MR 2029471 (2005d:37007)
10.: M. Einsiedler and T. Ward, Isomorphism rigidity in entropy rank two, Israel J. Math., vol. 147, 269-284, 2005. MR 2166364 (2006m:28022)
11.: E. Glasner, J.-P. Thouvenot and B. Weiss, Entropy theory without a past, Ergodic Theory Dynam. Systems, vol. 20, 1355-1370, 2000. MR 1786718 (2001h:37011)
12.: A. Katok, S. Katok and K. Schmidt, Rigidity of measurable structure for ${\mathbb{Z}}^d$ -actions by automorphisms of a torus, Comment. Math. Helv., vol. 77, 718-745, 2002. MR 1949111 (2003h:37007)
13.: B. Kitchens and K. Schmidt, Isomorphism rigidity of irreducible algebraic ${\mathbb{Z}}^{d}$ -actions, Invent. Math., vol. 142, 559-577, 2000. MR 1804161 (2001j:37004)
14.: D. Lind, The structure of skew products with ergodic group automorphisms, Israel J. Math., vol. 28, 205-248, 1977. MR 0460593 (57:586)
15.: D. Lind, K. Schmidt and T. Ward, Mahler measure and entropy for commuting automorphisms of compact groups, Invent. Math., vol. 101, 593-629, 1990. MR 1062797 (92j:22013)
16.: M. Misiurewicz, Topological conditional entropy, Studia Math., vol. 55, 175-200, 1976. MR 0415587 (54:3672)
17.: D. Rudolph and K. Schmidt, Almost block independence and Bernoullicity of ${\mathbb{Z}}^{d}$ -actions by automorphisms of compact abelian groups, Invent. Math., vol. 120, 455-488, 1995. MR 1334481 (96d:22004)
18.: K. Schmidt, Invariant measures for certain expansive ${\mathbb{Z}}^{2}$ -actions, Israel J. Math., vol. 90, 295-300, 1995. MR 1336327 (96c:28028)
19.: K. Schmidt, Dynamical systems of algebraic origin, Progress in Mathematics, 128, Birkhäuser Verlag, Basel, 1995. MR 1345152 (97c:28041)
20.: K. Schmidt, Algebraic ${\mathbb{Z}}^{d}$ -actions, Pacific Institute for the Mathematical Sciences Distinguished Chair Lecture Notes (electronic publication), University of Victoria, BC, 2002.
21.: K. Schmidt and T. Ward, Mixing automorphisms of compact groups and a theorem of Schlickewei, Invent. Math., vol. 111, 69-76, 1993. MR 1193598 (95c:22011)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|