Algebraic shift equivalence and primitive matrices

Boyle, Mike; Handelman, David

doi:10.1090/S0002-9947-1993-1102219-4

Algebraic shift equivalence and primitive matrices
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Trans. Amer. Math. Soc. 336 (1993), 121-149 Request permission

Abstract:

Motivated by symbolic dynamics, we study the problem, given a unital subring $S$ of the reals, when is a matrix $A$ algebraically shift equivalent over $S$ to a primitive matrix? We conjecture that simple necessary conditions on the nonzero spectrum of $A$ are sufficient, and establish the conjecture in many cases. If $S$ is the integers, we give some lower bounds on sizes of realizing primitive matrices. For Dedekind domains, we prove that algebraic shift equivalence implies algebraic strong shift equivalence.

References

Gert Almkvist, Endomorphisms of finitely generated projective modules over a commutative ring, Ark. Mat. 11 (1973), 263–301. MR 424786, DOI 10.1007/BF02388522
Jonathan Ashley, Resolving factor maps for shifts of finite type with equal entropy, Ergodic Theory Dynam. Systems 11 (1991), no. 2, 219–240. MR 1116638, DOI 10.1017/S0143385700006118

Algebraic

theory

Kirby A. Baker, Strong shift equivalence of $2\times 2$ matrices of nonnegative integers, Ergodic Theory Dynam. Systems 3 (1983), no. 4, 501–508. MR 753918, DOI 10.1017/S0143385700002091
Kirby A. Baker, Strong shift equivalence and shear adjacency of nonnegative square integer matrices, Linear Algebra Appl. 93 (1987), 131–147. MR 898550, DOI 10.1016/S0024-3795(87)90319-3
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. MR 912638, DOI 10.1090/memo/0377
Mike Boyle, Shift equivalence and the Jordan form away from zero, Ergodic Theory Dynam. Systems 4 (1984), no. 3, 367–379. MR 776874, DOI 10.1017/S0143385700002510
Mike Boyle and David Handelman, The spectra of nonnegative matrices via symbolic dynamics, Ann. of Math. (2) 133 (1991), no. 2, 249–316. MR 1097240, DOI 10.2307/2944339
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. MR 927684, DOI 10.1090/S0002-9947-1988-0927684-2

On Williams’ problem for positive matrices

George A. Elliott, On totally ordered groups, and $K_{0}$, Ring theory (Proc. Conf., Univ. Waterloo, Waterloo, 1978) Lecture Notes in Math., vol. 734, Springer, Berlin, 1979, pp. 1–49. MR 548122
Ulf-Rainer Fiebig, Gyration numbers for involutions of subshifts of finite type. I, Forum Math. 4 (1992), no. 1, 77–108. MR 1142474, DOI 10.1515/form.1992.4.77
David Handelman, Positive matrices and dimension groups affiliated to $C^{\ast }$-algebras and topological Markov chains, J. Operator Theory 6 (1981), no. 1, 55–74. MR 637001

Reducible toplogical Markov chains via

theory and Ext

David Handelman, Eventually positive matrices with rational eigenvectors, Ergodic Theory Dynam. Systems 7 (1987), no. 2, 193–196. MR 896791, DOI 10.1017/S014338570000393X

Strongly indecomposable abelian groups and totally ordered topological Markov chains

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. MR 564188
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. MR 837977, DOI 10.1017/S0143385700003308
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. MR 1130528, DOI 10.1090/S0894-0347-1992-1130528-4
K. H. Kim and F. W. Roush, Strong shift equivalence of Boolean and positive rational matrices, Linear Algebra Appl. 161 (1992), 153–164. MR 1142735, DOI 10.1016/0024-3795(92)90010-8
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. MR 1101086, DOI 10.1017/S0143385700006039
Wolfgang Krieger, On dimension functions and topological Markov chains, Invent. Math. 56 (1980), no. 3, 239–250. MR 561973, DOI 10.1007/BF01390047
Wolfgang Krieger, On the subsystems of topological Markov chains, Ergodic Theory Dynam. Systems 2 (1982), no. 2, 195–202 (1983). MR 693975, DOI 10.1017/S0143385700001516
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. MR 1101088, DOI 10.1017/S0143385700006052
John Milnor, Introduction to algebraic $K$-theory, Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. MR 0349811
Morris Newman, Integral matrices, Pure and Applied Mathematics, Vol. 45, Academic Press, New York-London, 1972. MR 0340283
William Parry and Selim Tuncel, On the stochastic and topological structure of Markov chains, Bull. London Math. Soc. 14 (1982), no. 1, 16–27. MR 642417, DOI 10.1112/blms/14.1.16
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. MR 466490, DOI 10.1112/plms/s3-35.3.483
Pierre Samuel, Théorie algébrique des nombres, Hermann, Paris, 1967 (French). MR 0215808
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. MR 0266227
R. F. Williams, Classification of subshifts of finite type, Ann. of Math. (2) 98 (1973), 120–153; errata, ibid. (2) 99 (1974), 380–381. MR 331436, DOI 10.2307/1970908

Strong shift equivalence of matrices in