Classification problems for shifts on modules over a principal ideal domain

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]  S. Mitter,et al.  Linear systems over Noetherian rings in the behavioural approach , 1994 .

[2]  M. Morf,et al.  New results in 2-D systems theory, part II: 2-D state-space models—Realization and the notions of controllability, observability, and minimality , 1977, Proceedings of the IEEE.

[3]  Eduardo Sontag.  Linear Systems over Commutative Rings. A Survey , 1976 .

[4]  F. Blanchard.  Systèmes dynamiques topologiques associés à des automates récurrents , 1981 .

[5]  Tsit Yuen Lam,et al.  Serre's Conjecture , 1978 .

[6]  D. Zelinsky.  Linearly compact modules and rings , 1953 .

[7]  Bruce Kitchens,et al.  Expansive dynamics on zero-dimensional groups , 1987, Ergodic Theory and Dynamical Systems.

[8]  Jan C. Willems,et al.  Models for Dynamics , 1989 .

[9]  Klaus Schmidt,et al.  Automorphisms of compact groups , 1989, Ergodic Theory and Dynamical Systems.

[10]  P. Fuhrmann.  Linear Systems and Operators in Hilbert Space , 1982 .

[11]  Georges Hansel,et al.  Systèmes Codés , 1986, Theor. Comput. Sci..

[12]  Karl Petersen.  Chains, entropy, coding , 1986 .

[13]  I. G. MacDonald.  Duality over complete local rings , 1962 .

[14]  John W. Bunce,et al.  Linear Systems over Commutative Rings , 1986 .

[15]  J. Leigh.  LINEAR SYSTEMS AND OPERATORS IN HILBERT SPACE , 1982 .

[16]  Fabio Fagnani.  Some results on the classification of expansive automorphisms of compact abelian groups , 1996 .

[17]  F. Fagnani.  Shifts on compact and discrete Lie groups:algebraic-topological invariants and classification problems , 1997 .

[18]  I. Salama.  Topological entropy and recurrence of countable chains. , 1988 .

[19]  Mitchell D. Trott,et al.  The dynamics of group codes: State spaces, trellis diagrams, and canonical encoders , 1993, IEEE Trans. Inf. Theory.

[20]  Klaus Schmidt,et al.  Automorphisms of Compact Abelian Groups and Affine Varieties , 1990 .

[21]  Sun-Yuan Kung,et al.  New Results in 2-D Systems Theory. Part III. Recursive Realization and Estimation Algorithms for 2-D Systems. , 1977 .