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Automorphisms of \(\mathbb Z^ d\)-subshifts of finite type. (English) Zbl 0823.28007

Let \((\Sigma, \sigma)\) be a \(\mathbb Z^ d\)-subshift of finite type. The automorphism group \(\operatorname{Aut}(\Sigma)\) is the group of homeomorphisms of \(\Sigma\) commuting with \(\sigma\). This paper extends to dimensions \(d>1\) results of M. Boyle, D. Lind and D. Rudolph [Trans. Am. Math. Soc. 306, No. 1, 71–114 (1988; Zbl 0664.28006)] and Curtis, Hedlund and Lyndon [see G. A. Hedlund, Topological Dynamics, Int. Symp. Colorado State Univ. 1967, 259–289 (1968; Zbl 0195.52702)]. In particular, it is shown that under a strong irreducibility condition, \(\operatorname{Aut}(\Sigma)\) contains any finite group. For \(\mathbb Z^ d\)-subshifts of finite type without this irreducibility condition, it is shown using examples that topological mixing is not sufficient to give any finite group in the automorphism group, in general. \(\operatorname{Aut}(\Sigma)\) is explicitly calculated for a particular topological mixing \(\mathbb Z^ 2\)-subshift of finite type and some open questions are posed.

MSC:

37B99 Topological dynamics
37A25 Ergodicity, mixing, rates of mixing
28D15 General groups of measure-preserving transformations
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[1] Berger, R., The undecidability of the Domino problem, Mem. Amer. Math. Soc., 66 (1966) · Zbl 0199.30802
[2] Boyle, M.; Lind, D.; Rudolph, D., The automorphism group of a shift of finite type, Trans. Amer. Math. Soc., 306, 71-114 (1988) · Zbl 0664.28006
[3] Burton, R. and J. Steif - Nonuniqueness of measures of maximal entropy for subshifts of finite type. Ergod. Th. and Dyn. Sys., to appear.; Burton, R. and J. Steif - Nonuniqueness of measures of maximal entropy for subshifts of finite type. Ergod. Th. and Dyn. Sys., to appear.
[4] Elsanousi, S. A., A variational principle for the pressure of a continuous \(Z^2\)-action on a compact metric space, Amer. J. Math., 99, 77-106 (1977) · Zbl 0388.28021
[5] Hedlund, G. A., Transformations commuting with the shift, (Topological Dynamics (1968), Benjamin: Benjamin New York), 259-289 · Zbl 0195.52702
[6] Hedlund, G. A., Endomorphisms and automorphisms of the shift dynamical system, Math. Sys. Theory, 3, 320-375 (1969) · Zbl 0182.56901
[7] Kim, K. H.; Roush, F. W.; Wagoner, J. B., Automorphisms of the dimension group and gyration numbers, J. Amer. Math. Soc., 5, 191-212 (1992) · Zbl 0749.54012
[8] Kitchens, B.; Schmidt, K., Markov subgroups of \((Z/2Z^{Z^2})\), Cont. Math., 135, 265-283 (1992) · Zbl 0774.58021
[9] Ledrappier, F., Un champ markovien peut être d’entropie nulle et melangeant, Comptes Rendus Acad. Sci. Paris Ser. A, 287, 561-562 (1978) · Zbl 0387.60084
[10] Lind, D. A., Entropies of automorphisms of a topological Markov shift, Proc. Amer. Math. Soc., 99, 589-595 (1987) · Zbl 0619.54030
[11] Misiurewicz, M., A short proof of the variational principle for a \(Z_+^n\)-action on a compact space, Astérisque, 49, 61-74 (1977) · Zbl 0351.54036
[12] Robinson, R., Undecidability and nonperiodicity for tilings of the plane, Invent. Math., 12, 177-209 (1971) · Zbl 0197.46801
[13] Ryan, J. P., The shift and commutivity, Math. Sys. Th., 6, 82-85 (1971)
[14] Ryan, J. P., The shift and commutivity II, Math. Sys. Th., 8, 249-250 (1975) · Zbl 0315.54051
[15] Schmidt, K., Algebraic ideas in ergodic theory, C.B.M.S. Reg. Conf. Ser. in Math., 76 (1990)
[16] Shereshevsky, M. A., On the classification of some two-dimensional Markov shifts with group structure, Ergod. Th. and Dyn. Sys., 12, 823-833 (1992) · Zbl 0781.58017
[17] Shereshevsky, M. A., Expansiveness, entropy and polynomial growth for groups acting on subshifts by automorphisms, Indag. Mathem., 4, 203-210 (1993) · Zbl 0794.28010
[18] Ward, T., An algebraic obstruction to isomorphism of Markov shifts with group alphabets, Bull. London Math. Soc., 25, 240-246 (1993) · Zbl 0792.22004
[19] Ward, T., Endomorphisms and automorphisms of group subshifts of finite type: non-abelian examples (1993), Preprint
[20] Ward, T., Entropy bounds for automorphisms commuting with algebraic actions (1993), Preprint
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