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Entropies of automorphisms of a topological Markov shift. (English) Zbl 0619.54030
Let $$\sigma$$ be a mixing topological Markov shift, $$\lambda$$ a weak Perron number, q(t) a polynomial with nonnegative integer coefficients, and r a nonnegative rational. We construct a homeomorphism commuting with $$\sigma$$ whose topological entropy is $$\log [q(\lambda)q(1/\lambda)]^ r$$. These values are shown to include the logarithms of all weak Perron numbers, and are dense in the nonnegative reals.

##### MSC:
 54H20 Topological dynamics (MSC2010) 54C70 Entropy in general topology 37A99 Ergodic theory
##### Keywords:
mixing topological Markov shift; weak Perron number
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##### References:
 [1] Mike Boyle and Wolfgang Krieger, Periodic points and automorphisms of the shift, Trans. Amer. Math. Soc. 302 (1987), no. 1, 125 – 149. · Zbl 0621.58031 [2] M. Boyle, D. Lind and D. Rudolph, The automorphism group of a subshift of finite type, preprint, Universities of Maryland and Washington, 1986. [3] Ethan M. Coven, Topological entropy of block maps, Proc. Amer. Math. Soc. 78 (1980), no. 4, 590 – 594. · Zbl 0452.54038 [4] Manfred Denker, Christian Grillenberger, and Karl Sigmund, Ergodic theory on compact spaces, Lecture Notes in Mathematics, Vol. 527, Springer-Verlag, Berlin-New York, 1976. · Zbl 0328.28008 [5] G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory 3 (1969), 320 – 375. · Zbl 0182.56901 · doi:10.1007/BF01691062 · doi.org [6] D. A. Lind, The entropies of topological Markov shifts and a related class of algebraic integers, Ergodic Theory Dynam. Systems 4 (1984), no. 2, 283 – 300. · Zbl 0546.58035 · doi:10.1017/S0143385700002443 · doi.org [7] Brian Marcus and Sheldon Newhouse, Measures of maximal entropy for a class of skew products, Ergodic theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978), Lecture Notes in Math., vol. 729, Springer, Berlin, 1979, pp. 105 – 125. · Zbl 0429.28020 [8] J. Patrick Ryan, The shift and commutativity, Math. Systems Theory 6 (1972), 82 – 85. · Zbl 0227.54037 · doi:10.1007/BF01706077 · doi.org [9] J. Wagoner, Markov partitions and $${K_2}$$, preprint, University of California, Berkeley, 1985. [10] 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 · doi.org [11] Stephen Wolfram, Universality and complexity in cellular automata, Phys. D 10 (1984), no. 1-2, 1 – 35. Cellular automata (Los Alamos, N.M., 1983). · doi:10.1016/0167-2789(84)90245-8 · doi.org
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