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$\mathbb {Z}^d$ group shifts and Bernoulli factors

Published online by Cambridge University Press:  01 April 2008

MIKE BOYLE and
MICHAEL SCHRAUDNER
MIKE BOYLE
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA (email: mmb@math.umd.edu)Centro de Modelamiento Matemático, Universidad de Chile, Av. Blanco Encalada 2120, Piso 7, Santiago de Chile, Chile (email: mschraudner@dim.uchile.cl)
MICHAEL SCHRAUDNER
Affiliation:
Centro de Modelamiento Matemático, Universidad de Chile, Av. Blanco Encalada 2120, Piso 7, Santiago de Chile, Chile (email: mschraudner@dim.uchile.cl)
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Abstract

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In this paper, a group shift is an expansive action of $\Z ^d$ on a compact metrizable zero-dimensional group by continuous automorphisms. All group shifts factor topologically onto equal-entropy Bernoulli shifts; abelian group shifts factor by continuous group homomorphisms onto canonical equal-entropy Bernoulli group shifts; and completely positive entropy abelian group shifts are weakly algebraically equivalent to these Bernoulli factors. A completely positive entropy group (even vector) shift need not be topologically conjugate to a Bernoulli shift, and the Pinsker factor of a vector shift need not split topologically.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 2: William Parry Memorial Volume , April 2008, pp. 367 - 387
Copyright
Copyright © Cambridge University Press 2008

References

[1]Boyle, M.. Factoring factor maps. J. London Math. Soc. (2) 57(2) (1998), 491–502.CrossRefGoogle Scholar
[2]Boyle, M., Lind, D. and Rudolph, D.. The automorphism group of a shift of finite type. Trans. Amer. Math. Soc. 306(1) (1988), 71–114.CrossRefGoogle Scholar
[3]Boyle, M. and Schraudner, M.. $\Z \sp d$ shifts of finite type without equal-entropy Bernoulli factors. J. Difference Equ. Appl. Special issue on Combinatorial and Topological Dynamics to appear.Google Scholar
[4]Einsiedler, M. and Schmidt, K.. The adjoint action of an expansive algebraic $\Z \sp d$ action. Monatsh. Math. 135(3) (2002), 203–220.CrossRefGoogle Scholar
[5]Fagnani, F.. Some results on the classification of expansive automorphisms of compact abelian groups. Ergod. Th. & Dynam. Sys. 16(1) (1996), 45–50.CrossRefGoogle Scholar
[6]Fagnani, F.. Shifts on compact and discrete Lie groups: algebraic-topological invariants and classification problems. Adv. Math. 127(2) (1997), 283–306.CrossRefGoogle Scholar
[7]Hochman, M. and Meyerovitch, T.. A characterization of the entropies of multidimensional shifts of finite type. Mathematics (2007). ArXiv DS/0703206.Google Scholar
[8]Johnson, A., Kass, S. and Madden, K.. Projectional entropy in higher dimensional shifts of finite type. Complex Systems to appear.Google Scholar
[9]Kitchens, B.. Expansive dynamics on zero-dimensional groups. Ergod. Th. & Dynam. Sys. 7 (1987), 249–261.CrossRefGoogle Scholar
[10]Kitchens, B.. Multidimensional convolutional codes. SIAM J. Discrete Math. 15(3) (2002), 367–381.CrossRefGoogle Scholar
[11]Kitchens, B.. The structure of d-dimensional vector shifts and convolutional codes, 2003, 18 pp., unpublished manuscript.Google Scholar
[12]Kitchens, B. and Schmidt, K.. Automorphisms of compact groups. Ergod. Th. & Dynam. Sys. 9 (1989), 691–735.CrossRefGoogle Scholar
[13]Lind, D. and Schmidt, K.. Homoclinic points of algebraic $\Z \sp d$-actions. J. Amer. Math. Soc. 12(4) (1999), 953–980.CrossRefGoogle Scholar
[14]Lind, D., Schmidt, K. and Ward, T.. Mahler measure and entropy for commuting automorphisms of compact groups. Invent. Math. 101 (1990), 593–629.CrossRefGoogle Scholar
[15]Ornstein, D. and Weiss, B.. Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48 (1987), 1–141.CrossRefGoogle Scholar
[16]Parry, W.. Intrinsic Markov chains. Trans. Amer. Math. Soc. 112 (1964), 55–66.CrossRefGoogle Scholar
[17]Rudolph, D. J. and Schmidt, K.. Almost block independence and Bernoullicity of $\Z \sp d$-actions by automorphisms of compact abelian groups. Invent. Math. 120(3) (1995), 455–488.CrossRefGoogle Scholar
[18]Rudolph, D. and Weiss, B.. Entropy and mixing for amenable group actions. Ann. Math. (2) 151(3) (2000), 1119–1150.CrossRefGoogle Scholar
[19]Schmidt, K.. Dynamical Systems of Algebraic Origin. Birkhäuser, Basel, 1995.Google Scholar