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The entropies of topological Markov shifts and a related class of algebraic integers

  • D. A. Lind (a1)
Abstract
Abstract

We give an algebraic characterization of the class of spectral radii of aperiodic non-negative integral matrices, and describe a method of constructing all such matrices with given spectral radius. The logarithms of the numbers in are the entropies of mixing topological Markov shifts. There is an arithmetic structure to , including factorization into irreducibles in only finitely many ways. This arithmetic structure has dynamical consequences, such as the impossibility of factoring the p-shift into a direct product of nontrivial homeomorphisms for prime p.

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[3]R. Bowen , Topological entropy and Axiom A. In Proc. Symp. Pure Math., 14, 2341, Amer. Math. Soc: Providence, RI., 1970.

[5]R. Bowen & O. E. Lanford III. Zeta functions of restrictions of the shift transformation. In Proc. Symp. Pure Math. 14, 4350, Amer. Math. Soc: Providence, R.I., 1970.

[12]W. Krieger . On dimension functions and topological Markov chains. Invent. Math. 56 (1980), 239250.

[16]W. Parry . Intrinsic Markov chains. Trans. Amer. Math. Soc. 112 (1964), 5556.

[18]Ya. Sinai . Markov partitions and C-diffeomorphisms. Funct. Anal. and its Appl., 2 (1968), No. 1, 6489.

[19]R. F. Williams . Classification of subshifts of finite type. Ann. of Math. 98 (1973), 120153; Errata, 99 (1974), 380–381.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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