Classification problems for shifts on modules over a principal ideal domain

Fagnani, Fabio; Zampieri, Sandro

doi:10.1090/S0002-9947-97-01876-X

Classification problems for shifts on modules over a principal ideal domain
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by Fabio Fagnani and Sandro Zampieri PDF

Trans. Amer. Math. Soc. 349 (1997), 1993-2006 Request permission

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.

References

F. Blanchard, Systèmes dynamiques topologiques associés à des automates récurrents, Z. Wahrsch. Verw. Gebiete 58 (1981), no. 4, 549–564 (French, with English summary). MR 638035, DOI 10.1007/BF00534947
James W. Brewer, John W. Bunce, and F. S. Van Vleck, Linear systems over commutative rings, Lecture Notes in Pure and Applied Mathematics, vol. 104, Marcel Dekker, Inc., New York, 1986. MR 839186
W. Brewer, J.W. Bunce, and F.S. Van Vleck. Linear systems over commutative rings. Dekker, 1986.
E. I. Dinaburg, An example of the computation of topological entropy, Uspehi Mat. Nauk 23 (1968), no. 4 (142), 249–250 (Russian). MR 0230321
F. Fagnani. Some results on the classification of expansive automorphisms of compact Abelian groups. Ergodic Theory and Dynamical Systems, 16 (1996), 45-50.
F. Fagnani. Shifts on compact and discrete Lie groups: algebraic-topological invariants and classification problems. To appear in Advances in Mathematics.
G. David Forney Jr. and Mitchell D. Trott, The dynamics of group codes: state spaces, trellis diagrams, and canonical encoders, IEEE Trans. Inform. Theory 39 (1993), no. 5, 1491–1513. MR 1281706, DOI 10.1109/18.259635
Paul A. Fuhrmann, Linear systems and operators in Hilbert space, McGraw-Hill International Book Co., New York, 1981. MR 629828
Bruce P. Kitchens, Expansive dynamics on zero-dimensional groups, Ergodic Theory Dynam. Systems 7 (1987), no. 2, 249–261. MR 896796, DOI 10.1017/S0143385700003989
Bruce Kitchens and Klaus Schmidt, Automorphisms of compact groups, Ergodic Theory Dynam. Systems 9 (1989), no. 4, 691–735. MR 1036904, DOI 10.1017/S0143385700005290
Wolfgang Krieger, On the uniqueness of the equilibrium state, Math. Systems Theory 8 (1974/75), no. 2, 97–104. MR 399412, DOI 10.1007/BF01762180
S. Kung, B. Lévy, M. Morf, and T. Kailath. New results in 2D systems theory. Part 1. Proc. IEEE, 65 (1977), 861-872.
T. Y. Lam, Serre’s conjecture, Lecture Notes in Mathematics, Vol. 635, Springer-Verlag, Berlin-New York, 1978. MR 0485842, DOI 10.1007/BFb0068340
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.
I. G. Macdonald, Duality over complete local rings, Topology 1 (1962), 213–235. MR 151491, DOI 10.1016/0040-9383(62)90104-0
G. Miles and R. K. Thomas, The breakdown of automorphisms of compact topological groups, Studies in probability and ergodic theory, Adv. in Math. Suppl. Stud., vol. 2, Academic Press, New York-London, 1978, pp. 207–218. MR 517262
Karl Petersen, Chains, entropy, coding, Ergodic Theory Dynam. Systems 6 (1986), no. 3, 415–448. MR 863204, DOI 10.1017/S014338570000359X
Ibrahim A. Salama, Topological entropy and recurrence of countable chains, Pacific J. Math. 134 (1988), no. 2, 325–341. MR 961239, DOI 10.2140/pjm.1988.134.325
Klaus Schmidt, Automorphisms of compact abelian groups and affine varieties, Proc. London Math. Soc. (3) 61 (1990), no. 3, 480–496. MR 1069512, DOI 10.1112/plms/s3-61.3.480
E.D. Sontag. Linear systems over commutative rings: A survey. Ricerche di Automatica, 7 (1976), 1-34.
Jan C. Willems, Models for dynamics, Dynamics reported, Vol. 2, Dynam. Report. Ser. Dynam. Systems Appl., vol. 2, Wiley, Chichester, 1989, pp. 171–269. MR 1000978
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.
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14