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Strong shift equivalence of 2 × 2 matrices of non-negative integers

Published online by Cambridge University Press:  19 September 2008

Kirby A. Baker
Kirby A. Baker
Affiliation:
Department of Mathematics, UCLA, California, CA 90024, USA
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Abstract

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The concept of strong shift equivalence of square non-negative integral matrices has been used by R. F. Williams to characterize topological isomorphism of the associated topological Markov chains. However, not much has been known about sufficient conditions for strong shift equivalence even for 2×2 matrices (other than those of unit determinant). The main theorem of this paper is: If A and B are positive 2×2 integral matrices of non-negative determinant and are similar over the integers, then A and B are strongly shift equivalent.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 3 , Issue 4 , December 1983 , pp. 501 - 508
Copyright
Copyright © Cambridge University Press 1983

References

REFERENCES

[1]Bondy, J. A. & Murty, U. S. R.. Graph Theory with Applications. American Elsevier: New York, 1976.CrossRefGoogle Scholar
[2]Borevich, Z. I. & Shafarevich, I. R.. Number Theory. Academic Press: New York, 1966.Google Scholar
[3]Bowen, R.. Markov partitions for axiom-A diffeomorphisms, Amer. J. Math. 92 (1970), 725–747.CrossRefGoogle Scholar
[4]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics Vol. 470. Springer-Verlag; New York, 1975.Google Scholar
[5]Bowen, R. & Lanford, O.. Zeta functions of restrictions of the shift transformation. In Proc. Symp. Pure Math. 14, pp. 43–50. Amer. Math. Soc.: Providence, R.I. 1970.Google Scholar
[6]Brocot, A.. Calcul des rouages par approximation. Paris, 1862;Google Scholar
(referenced in: Dixon, L. E., History of the Theory of Numbers, Vol. I, p. 156. Chelsea: New York, 1952).Google Scholar
[7]Cuntz, J. & Krieger, W.. Topological Markov chains with dicyclic dimension groups. J. Reine Angew. Math. 320 (1980), 44–51.Google Scholar
[8]Denker, M., Grillenberger, C. & Sigmund, K.. Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics, Vol. 527. Springer-Verlag: New York 1976.Google Scholar
[9]Effros, E.. Dimensions and C*-algebras, CBMS Regional Conference Series, No. 46. Amer. Math. Soc.: Providence, 1981.CrossRefGoogle Scholar
[10]Effros, E.. On Williams' problem for positive matrices. Preprint, 1981.Google Scholar
[11]Effros, E. & Shen, C.-D., Approximately finite C*-algebras and continued fractions. Indiana J. Math. 29 (1980), 191–204.Google Scholar
[12]Gantmacher, F. R.. The Theory of Matrices, Vol. II. Chelsea Publ. Co.: New York, 1960.Google Scholar
[13]Handelman, D.. Positive matrices and dimension groups affiliated to C*-algebras and topological Markov chains, J. Operator Theory 6 (1981), 55–74.Google Scholar
[14]Handelman, D.. Strongly indecomposable abelian groups and totally ordered topological Markov chains. Preprint, 1981.Google Scholar
[15]Hardy, G. H. & Wright, E. M.. An Introduction to the Theory of Numbers, 4th. ed.Oxford U. Press: New York, 1960.Google Scholar
[16]Kim, K. H. & Rousch, F. W.. Some results on decidability of shift equivalence, J. Combin. Inform. Systems Sci. 4 (1979), 123–146.Google Scholar
[17]Krieger, W.. On dimension functions and topological Markov chains, Invent. Math. 63 (1980), 239–250.CrossRefGoogle Scholar
[18]Latimer, C. G. & MacDuffee, C. C.. A correspondence between classes of ideals and classes of matrices. Ann. of Math. 34 (1933), 313–316.Google Scholar
[19]Morse, M. & Hedlund, G.. Symbolic dynamics. Amer. J. Math. 60 (1938), 815–866.Google Scholar
[20]Newman, M.. Integral Matrices. Academic Press: New York, 1972.Google Scholar
[21]Parry, W.. Intrinsic Markov chains. Trans. Amer. Math. Soc. 112 (1964), 55–66.CrossRefGoogle Scholar
[22]Parry, W.. The classification of topological Markov chains. Adapted shift equivalence. Israel J. Math. 38 (1981), 335–344.Google Scholar
[23]Parry, W. & Williams, R. F.. Block coding and a zeta function for finite Markov chains. Proc. London Math. Soc. 35 (1977), 484–495.Google Scholar
[24]Rademacher, H.. Lectures on Elementary Number Theory. Blaisdell: New York, 1964.Google Scholar
[25]Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.Google Scholar
[26]Taussky, O.. On a theorem of Latimer and McDuffee. Canad. J. Math. 1 (1949), 300–302.Google Scholar
[27]Williams, R. F.. Classification of one dimensional attractors. In Proc. Sympos. Pure Math. Vol. 14, pp. 341–361. Amer. Math. Soc.: Providence, 1970.Google Scholar
[28]Williams, R. F.. Classification of subshifts of finite type. Ann. of Math. 98 (1973), 120–153;CrossRefGoogle Scholar
Errata, Ann. of Math. 99 (1974), 380–381.Google Scholar