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Algebraic shift equivalence and primitive matrices. (English) Zbl 0766.15024

In an earlier paper the authors obtained sufficient conditions for a characteristic polynomial to be realizable up to a factor \(t^ n\) by a nonnegative primitive matrix over \(Z\) or more general subrings of the reals.
Here they obtain sufficient conditions to realize it within a given shift equivalence class; if the conditions of their spectral conjecture hold, the eigenvalues are from the subring which is \(Z\) or a Dedekind domain with a nontrivial unit, this can be done.
They also obtain results on shift equivalence over rings, matrices with 2 nonzero integer eigenvalues, and bound on the size of a realization.

MSC:

15B48 Positive matrices and their generalizations; cones of matrices
15A21 Canonical forms, reductions, classification
15B33 Matrices over special rings (quaternions, finite fields, etc.)
06F25 Ordered rings, algebras, modules
37E99 Low-dimensional dynamical systems
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[1] Gert Almkvist, Endomorphisms of finitely generated projective modules over a commutative ring, Ark. Mat. 11 (1973), 263 – 301. · Zbl 0278.13005 · doi:10.1007/BF02388522
[2] Jonathan Ashley, Resolving factor maps for shifts of finite type with equal entropy, Ergodic Theory Dynam. Systems 11 (1991), no. 2, 219 – 240. · Zbl 0741.54014 · doi:10.1017/S0143385700006118
[3] H. Bass, Algebraic \( K\)-theory, Benjamin, New York, 1973. · Zbl 0265.00008
[4] Kirby A. Baker, Strong shift equivalence of 2\times 2 matrices of nonnegative integers, Ergodic Theory Dynam. Systems 3 (1983), no. 4, 501 – 508. · Zbl 0544.54035 · doi:10.1017/S0143385700002091
[5] Kirby A. Baker, Strong shift equivalence and shear adjacency of nonnegative square integer matrices, Linear Algebra Appl. 93 (1987), 131 – 147. · Zbl 0621.15011 · doi:10.1016/S0024-3795(87)90319-3
[6] Mike Boyle, Brian Marcus, and Paul Trow, Resolving maps and the dimension group for shifts of finite type, Mem. Amer. Math. Soc. 70 (1987), no. 377, vi+146. · Zbl 0651.54018 · doi:10.1090/memo/0377
[7] Mike Boyle, Shift equivalence and the Jordan form away from zero, Ergodic Theory Dynam. Systems 4 (1984), no. 3, 367 – 379. · Zbl 0553.15006 · doi:10.1017/S0143385700002510
[8] Mike Boyle and David Handelman, The spectra of nonnegative matrices via symbolic dynamics, Ann. of Math. (2) 133 (1991), no. 2, 249 – 316. · Zbl 0735.15005 · doi:10.2307/2944339
[9] Mike Boyle, Douglas Lind, and Daniel Rudolph, The automorphism group of a shift of finite type, Trans. Amer. Math. Soc. 306 (1988), no. 1, 71 – 114. · Zbl 0664.28006
[10] E. G. Effros, On Williams’ problem for positive matrices, unpublished manuscript, 1981.
[11] George A. Elliott, On totally ordered groups, and \?\(_{0}\), Ring theory (Proc. Conf., Univ. Waterloo, Waterloo, 1978) Lecture Notes in Math., vol. 734, Springer, Berlin, 1979, pp. 1 – 49.
[12] Ulf-Rainer Fiebig, Gyration numbers for involutions of subshifts of finite type. I, Forum Math. 4 (1992), no. 1, 77 – 108. · Zbl 0751.58029 · doi:10.1515/form.1992.4.77
[13] David Handelman, Positive matrices and dimension groups affiliated to \?*-algebras and topological Markov chains, J. Operator Theory 6 (1981), no. 1, 55 – 74. · Zbl 0495.06011
[14] -, Reducible toplogical Markov chains via \( {K_0}\)-theory and Ext, Contemp. Math., vol. 10, Amer. Math. Soc., Providence, R.I., 1982, pp. 41-76.
[15] David Handelman, Eventually positive matrices with rational eigenvectors, Ergodic Theory Dynam. Systems 7 (1987), no. 2, 193 – 196. · Zbl 0629.15013 · doi:10.1017/S014338570000393X
[16] -, Strongly indecomposable abelian groups and totally ordered topological Markov chains, unpublished manuscript, 1981.
[17] K. H. Kim and F. W. Roush, Some results on decidability of shift equivalence, J. Combin. Inform. System Sci. 4 (1979), no. 2, 123 – 146. · Zbl 0438.15020
[18] Ki Hang Kim and Fred W. Roush, On strong shift equivalence over a Boolean semiring, Ergodic Theory Dynam. Systems 6 (1986), no. 1, 81 – 97. · Zbl 0574.15005 · doi:10.1017/S0143385700003308
[19] K. H. Kim and F. W. Roush, Williams’s conjecture is false for reducible subshifts, J. Amer. Math. Soc. 5 (1992), no. 1, 213 – 215. · Zbl 0749.54013
[20] K. H. Kim and F. W. Roush, Strong shift equivalence of Boolean and positive rational matrices, Linear Algebra Appl. 161 (1992), 153 – 164. · Zbl 0744.15012 · doi:10.1016/0024-3795(92)90010-8
[21] Bruce Kitchens, Brian Marcus, and Paul Trow, Eventual factor maps and compositions of closing maps, Ergodic Theory Dynam. Systems 11 (1991), no. 1, 85 – 113. · Zbl 0703.54023 · doi:10.1017/S0143385700006039
[22] Wolfgang Krieger, On dimension functions and topological Markov chains, Invent. Math. 56 (1980), no. 3, 239 – 250. · Zbl 0431.54024 · doi:10.1007/BF01390047
[23] Wolfgang Krieger, On the subsystems of topological Markov chains, Ergodic Theory Dynam. Systems 2 (1982), no. 2, 195 – 202 (1983). · Zbl 0508.54032 · doi:10.1017/S0143385700001516
[24] Brian Marcus and Selim Tuncel, The weight-per-symbol polytope and scaffolds of invariants associated with Markov chains, Ergodic Theory Dynam. Systems 11 (1991), no. 1, 129 – 180. · Zbl 0725.60071 · doi:10.1017/S0143385700006052
[25] John Milnor, Introduction to algebraic \?-theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. Annals of Mathematics Studies, No. 72. · Zbl 0237.18005
[26] Morris Newman, Integral matrices, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 45. · Zbl 0254.15009
[27] William Parry and Selim Tuncel, On the stochastic and topological structure of Markov chains, Bull. London Math. Soc. 14 (1982), no. 1, 16 – 27. · Zbl 0481.28015 · doi:10.1112/blms/14.1.16
[28] William Parry and R. F. Williams, Block coding and a zeta function for finite Markov chains, Proc. London Math. Soc. (3) 35 (1977), no. 3, 483 – 495. · Zbl 0383.94011 · doi:10.1112/plms/s3-35.3.483
[29] Pierre Samuel, Théorie algébrique des nombres, Hermann, Paris, 1967 (French). · Zbl 0239.12001
[30] R. F. Williams, Classification of one dimensional attractors, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 341 – 361.
[31] R. F. Williams, Classification of subshifts of finite type, Ann. of Math. (2) 98 (1973), 120 – 153; errata, ibid. (2) 99 (1974), 380 – 381. · Zbl 0282.58008 · doi:10.2307/1970908
[32] -, Strong shift equivalence of matrices in \( {\text{GL}}(2,{\mathbf{Z}})\), unpublished manuscript, 1981.
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