Transactions of the American Mathematical Society

Author(s): Fabio Fagnani; Sandro Zampieri
Journal: Trans. Amer. Math. Soc. 349 (1997), 1993-2006.
MSC (1991): Primary 54C70, 15A23, 93B25
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract: In this paper we study symbolic dynamics over alphabets which are modules over a principal ideal domain, considering topological shifts which are also submodules. Our main result is the classification, up to algebraic and topological conjugacy, of the torsion-free, transitive, finite memory shifts.

1.: F. Blanchard. Systèmes dynamiques topologiques associés à des automates récurrents. Z. Wahr. Verw. Geb., 58 (1981), 549-564. MR 83c:68101
2.: F. Blanchard, G. Hansel. Systèmes codés. Theoretical Computer Science, 44 (1986), 17-49. MR 87i:93002
3.: W. Brewer, J.W. Bunce, and F.S. Van Vleck. Linear systems over commutative rings. Dekker, 1986.
4.: E.E. Dinaburg. An example of the computation of topological entropy. Uspehi Mat. Nauk., 23 (1968), 249-250. MR 37:5883
5.: F. Fagnani. Some results on the classification of expansive automorphisms of compact Abelian groups. Ergodic Theory and Dynamical Systems, 16 (1996), 45-50. CMP 96:8
6.: F. Fagnani. Shifts on compact and discrete Lie groups: algebraic-topological invariants and classification problems. To appear in Advances in Mathematics.
7.: G.D. Forney and M.D. Trott. The dynamics of group codes: State spaces, trellis diagrams and canonical encoders. IEEE Trans. on Information Theory, IT-39 (1993), 1491-1513. MR 96e:94017
8.: P. Fuhrmann Linear systems and operators in Hilbert spaces. McGraw-Hill, New York, 1981. MR 83d:47001
9.: B.P. Kitchens. Expansive dynamics on zero-dimensional groups. Ergodic Theory and Dynamical Systems, 7 (1987), 249-261. MR 88i:28039
10.: B.P. Kitchens and K. Schmidt. Automorphisms of compact groups. Ergodic Theory and Dynamical Systems, 9 (1989), 691-735. MR 91g:22008
11.: W. Krieger. On the uniqueness of the equilibrium state. Math. Systems Theory, 8 (1974), 97-104. MR 53:3256
12.: S. Kung, B. Lévy, M. Morf, and T. Kailath. New results in 2D systems theory. Part 1. Proc. IEEE, 65 (1977), 861-872.
13.: T.Y Lam, Serre's conjecture Springer LNM 635, 1978. MR 58:5644
14.: B. Lévy. 2D polynomial and rational matrices, and their applications for the modelling of 2D dynamical systems. Technical Report M735-11, Stanford Electronics Lab., 1981.
15.: I.G. MacDonald. Duality over complete local rings. Topology, 1 (1962), 213-235. MR 27:1476
16.: G. Miles and R.K. Thomas. The breakdown of automorphisms of compact topological groups. In G.C. Rota, editor, Studies in Probability and Ergodic Theory, 207-218. Academic Press, 1978. MR 80c:22007
17.: K. Petersen. Chains, entropy, coding. Ergodic Theory Dynamical Systems, 6 (1986), 415-448. MR 88i:28040
18.: I.A. Salama. Topological entropy and recurrence of countable chains. Pacific J. Math., 134 (1988), 325-341; errata: Pacific J. Math., 140 (1989), 397-398. MR 90d:54076; MR 90k:54055
19.: K. Schmidt. Automorphisms of compact Abelian groups and affine varieties. Proc. London Math. Soc. 61 (1990), 480-496. MR 91j:28015
20.: E.D. Sontag. Linear systems over commutative rings: A survey. Ricerche di Automatica, 7 (1976), 1-34.
21.: J.C. Willems. Models for dynamics. Dynamics Reported, 2 (1988), 171-269. MR 90g:58121
22.: S. Zampieri and S.K. Mitter. Linear systems over Noetherian rings in the behavioural approach. Journal of Mathematical Systems, Estimation, and Control, 6 (1996), 235-238.
23.: D. Zelinsky. Linearly compact modules and rings. Amer. J. Math., 75 (1953), 79-90. MR 14:532a

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 54C70, 15A23, 93B25

Fabio Fagnani
Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
Email: fagnani@cibs.sns.it

Sandro Zampieri
Affiliation: Dipartimento di Elettronica ed Informatica, Università di Padova, via Gradenigo, 6/a, 35131 Padova, Italy
Email: zampi@paola.dei.unipd.it

DOI: 10.1090/S0002-9947-97-01876-X
PII: S 0002-9947(97)01876-X
Keywords: Shifts, dynamical systems, transitivity, principal ideal domains, polynomial matrices, behavioral approach
Received by editor(s): November 15, 1995
Copyright of article: Copyright 1997, 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