Transactions of the American Mathematical Society

Author(s): Manfred Einsiedler; Douglas Lind
Journal: Trans. Amer. Math. Soc. 356 (2004), 1799-1831.
MSC (2000): Primary 37A35, 37B40, 54H20; Secondary 37A45, 37D20, 13F20
Posted: January 6, 2004
Retrieve article in: PDF DVI PostScript

Abstract: We investigate algebraic $\mathbb Z^d$ -actions of entropy rank one, namely those for which each element has finite entropy. Such actions can be completely described in terms of diagonal actions on products of local fields using standard adelic machinery. This leads to numerous alternative characterizations of entropy rank one, both geometric and algebraic. We then compute the measure entropy of a class of skew products, where the fiber maps are elements from an algebraic $\mathbb Z^d$ -action of entropy rank one. This leads, via the relative variational principle, to a formula for the topological entropy of continuous skew products as the maximum of a finite number of topological pressures. We use this to settle a conjecture concerning the relational entropy of commuting toral automorphisms.

1.: L. M. Abramov and V. A. Rohlin, Entropy of a skew product of mappings with invariant measure, Vestnik Leningrad Univ. 17 (1962), 5-13; AMS Translations, Ser. 2, 48, 255-265. MR 25:4076
2.: Roy L. Adler, A note on the entropy of skew product transformations, Proc. Amer. Math. Soc. 14 (1963), 665-669. MR 27:3779
3.: Rufus Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401-414; erratum, 181 (1973), 509-510. MR 43:469
4.: Mike Boyle and Douglas Lind, Expansive subdynamics, Trans. Amer. Math. Soc. 349 (1997), 55-102. MR 97d:58115
5.: David Eisenbud, Commutative Algebra with a View toward Algebraic Geometry, Springer, New York, 1995. MR 97a:13001
6.: Manfred Einsiedler, Douglas Lind, Richard Miles, and Thomas Ward, Expansive subdynamics for algebraic $\mathbb Z^d$ -actions, Ergodic Th. & Dyn. Syst. 21 (2001), 1695-1729.
7.: Manfred Einsiedler and Klaus Schmidt, Irreducibility, homoclinic points, and adjoint actions of algebraic $\mathbb{Z} ^d$ -actions of rank one, preprint.
8.: Shmuel Friedland, Entropy of graphs, semigroups and groups, Proceedings of Warwick Symposium on $\mathbb{Z} ^d$ -Actions, LMS Lecture Notes Series 228 (1996), Cambridge Univ. Press, 319-343. MR 97f:58080
9.: W. Geller and M. Pollicott, An entropy for $\mathbb{Z} ^2$ -actions with finite entropy generators, Fundamenta Math. 157 (1998), 209-220. MR 99h:58105
10.: Nathan Jacobson, Basic Algebra II, Freeman, San Francisco, 1980. MR 81g:00001
11.: Steve Kalikow, ${T},\,{T}\sp{-1}$ transformation is not loosely Bernoulli, Annals of Math. 115 (1982), 393-409. MR 85j:28019
12.: Boris Kalinin and Anatole Katok, Invariant measures for actions of higher rank abelian groups, Proc. Symp. Pure Math. 69, Amer. Math. Soc., Providence, 2001. MR 2002i:37035
13.: Anatole Katok, Svetlana Katok, and Klaus Schmidt, Rigidity of measurable structure for algebraic actions of higher-rank abelian groups, ESI-Preprint: ftp://ftp.esi.ac.at/pub/Preprints/esi850.ps.
14.: Anatole Katok and Ralf Spatzier, Invariant measures for higher rank hyperbolic abelian actions, Ergod. Th. & Dyn. Syst. 16 (1996), 751-778; corrections, 18 (1998), 503-507. MR 97d:58116; MR 99c:58093
15.: Bruce Kitchens and Klaus Schmidt, Isomorphism rigidity for irreducible algebraic $\mathbb{Z} ^d$ -actions, Invent. Math. 142 (2000), 559-577. MR 2001j:37004
16.: Francois Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant, C. R. Acad. Sci. Paris, Ser. A 287 (1978), 561-563. MR 80b:28030
17.: Francois Ledrappier and Peter Walters, A relativised variational principle for continuous transformations, J. London Math. Soc. 16 (1977), 568-576. MR 57:16540
18.: D. A. Lind, Ergodic automorphisms of the infinite torus are Bernoulli, Israel J. Math. 17 (1974), 162-168. MR 49:10856
19.: Douglas Lind, Klaus Schmidt, and Thomas Ward, Mahler measure and entropy for commuting automorphisms of compact groups, Invent. Math. 101 (1990), 593-629. MR 92j:22013
20.: Douglas Lind and Thomas Ward, Automorphisms of solenoids and -adic entropy, Ergodic Th. & Dyn. Sys. 8 (1988), 411-419. MR 90a:28031
21.: Brian Marcus and Sheldon Newhouse, Measures of maximal entropy for a class of skew products, Springer Lecture Notes in Math. 729 (1978), pp. 105-125. MR 80j:28030
22.: John Milnor, On the entropy geometry of cellular automata, Complex Systems 2 (1988), 357-386. MR 90c:54026
23.: Kyewon Koh Park, On directional entropy functions, Israel J. Math. 113 (1999), 243-267. MR 2000m:37007
24.: K. R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York, 1967. MR 37:2271
25.: Dinakar Ramakrishnan and Robert J. Valenza, Fourier Analysis on Number Fields, Springer, New York, 1999. MR 2000d:11002
26.: Daniel Rudolph, $\times2$ and $\times3$ invariant measures and entropy, Ergodic Th. & Dynam. Syst. 10 (1990), 395-406. MR 91g:28026
27.: Klaus Schmidt, Automorphisms of compact abelian groups and affine varieties, Proc. London Math. Soc. (3), 61 (1990), 480-496. MR 91j:28015
28.: Klaus Schmidt, Dynamical Systems of Algebraic Origin, Birkhäuser Verlag, Basel-Berlin-Boston, 1995. MR 97c:28041
29.: Mark Shereshevsky, Expansiveness, entropy and polynomial growth for groups acting on subshifts by automorphisms, Indag. Math. (2)f 4 (1993), 203-210. MR 94c:54074
30.: Peter Walters, An Introduction to Ergodic Theory, Springer, New York, 1982. MR 84e:28017
31.: André Weil, Basic Number Theory (Third Edition), Springer, New York, 1974. MR 55:302

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37A35, 37B40, 54H20, 37A45, 37D20, 13F20

Manfred Einsiedler
Affiliation: Department of Mathematics, Pennsylvania State University, State College, Pennsylvania 16802
Address at time of publication: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Email: einsiedl@math.psu.edu, einsiedl@math.washington.edu

Douglas Lind
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Email: lind@math.washington.edu

DOI: 10.1090/S0002-9947-04-03554-8
PII: S 0002-9947(04)03554-8
Keywords: Entropy, skew product, algebraic action, variational principle
Received by editor(s): July 26, 2002
Posted: January 6, 2004
Additional Notes: The first author gratefully acknowledges the hospitality of the University of Washington and the Penn State University, and was supported by the FWF research project P14379-MAT and the Erwin Schrödinger Stipendium J2090
The second author thanks the generous hospitality of the Yale Mathematics Department
Copyright of article: Copyright 2004, American Mathematical Society

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

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS