An invariant for continuous factors of Markov shifts

Kitchens, Bruce

doi:10.1090/S0002-9939-1981-0630029-8

An invariant for continuous factors of Markov shifts
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Proc. Amer. Math. Soc. 83 (1981), 825-828 Request permission

Abstract:

Let ${\Sigma _A}$ and ${\Sigma _B}$ be subshifts of finite type with Markov measures $(p,P)$ and $(q,Q)$. It is shown that if there is a continuous onto measure-preserving factor map from ${\Sigma _A}$ to ${\Sigma _B}$, then the block of the Jordan form of $Q$ with nonzero eigenvalues is a principal submatrix of the Jordan form of $P$. If ${\Sigma _A}$ and ${\Sigma _B}$ are irreducible with the same topological entropy, then the same relationship holds for the matrices $A$ and $B$. As a consequence, ${\zeta _B}(t)/{\zeta _A}(t)$, the ratio of the zeta functions, is a polynomial. From this it is possible to construct a pair of equalentropy subshifts of finite type that have no common equal-entropy continuous factor of finite type, and a strictly sofic system that cannot have an equal-entropy subshift of finite type as a continuous factor.

References

Roy L. Adler and Brian Marcus, Topological entropy and equivalence of dynamical systems, Mem. Amer. Math. Soc. 20 (1979), no. 219, iv+84. MR 533691, DOI 10.1090/memo/0219
R. Bowen and O. E. Lanford III, Zeta functions of restrictions of the shift transformation, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 43–49. MR 0271401
Ethan M. Coven and Michael E. Paul, Endomorphisms of irreducible subshifts of finite type, Math. Systems Theory 8 (1974/75), no. 2, 167–175. MR 383378, DOI 10.1007/BF01762187
Ethan M. Coven and Michael E. Paul, Sofic systems, Israel J. Math. 20 (1975), no. 2, 165–177. MR 383379, DOI 10.1007/BF02757884
Joachim Cuntz and Wolfgang Krieger, Topological Markov chains with dicyclic dimension groups, J. Reine Angew. Math. 320 (1980), 44–51. MR 592141, DOI 10.1515/crll.1980.320.44
G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory 3 (1969), 320–375. MR 259881, DOI 10.1007/BF01691062
Brian Marcus, Factors and extensions of full shifts, Monatsh. Math. 88 (1979), no. 3, 239–247. MR 553733, DOI 10.1007/BF01295238
Masakazu Nasu, Uniformly finite-to-one and onto extensions of homomorphisms between strongly connected graphs, Discrete Math. 39 (1982), no. 2, 171–197. MR 675863, DOI 10.1016/0012-365X(82)90141-8
William Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc. 112 (1964), 55–66. MR 161372, DOI 10.1090/S0002-9947-1964-0161372-1
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
Benjamin Weiss, Subshifts of finite type and sofic systems, Monatsh. Math. 77 (1973), 462–474. MR 340556, DOI 10.1007/BF01295322
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