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2016, 10: 483-495. doi: 10.3934/jmd.2016.10.483

The automorphism group of a minimal shift of stretched exponential growth

Van Cyr 1, and  Bryna Kra 2,

1. 

Department of Mathematics, Bucknell University, 1 Dent Drive, Lewisburg, PA 17837, United States
2. 

Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, United States

Received  September 2015 Revised  August 2016 Published  October 2016

The group of automorphisms of a symbolic dynamical system is countable, but often very large. For example, for a mixing subshift of finite type, the automorphism group contains isomorphic copies of the free group on two generators and the direct sum of countably many copies of $\mathbb{Z}$. In contrast, the group of automorphisms of a symbolic system of zero entropy seems to be highly constrained. Our main result is that the automorphism group of any minimal subshift of stretched exponential growth with exponent $<1/2$, is amenable (as a countable discrete group). For shifts of polynomial growth, we further show that any finitely generated, torsion free subgroup of Aut(X) is virtually nilpotent.
Keywords: zero entropy, automorphism, Subshift, amenable., block complexity.
Mathematics Subject Classification: Primary: 37B10; Secondary: 43A07, 54H20, 68R1.
Citation: Van Cyr, Bryna Kra. The automorphism group of a minimal shift of stretched exponential growth. Journal of Modern Dynamics, 2016, 10: 483-495. doi: 10.3934/jmd.2016.10.483
References:
[1]

H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3), 25 (1972), 603-614.  Google Scholar

[2]

M. Boyle, D. Lind and D. Rudolph, The automorphism group of a shift of finite type, Trans. Amer. Math. Soc., 306 (1988), 71-114. doi: 10.1090/S0002-9947-1988-0927684-2.  Google Scholar

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E. Coven, A. Quas and R. Yassawi, Computing automorphism groups of shifts, using atypical equivalence classes, Discrete Anal., (2016), 1-28. doi: 10.19086/da.611.  Google Scholar

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V. Cyr and B. Kra, The automorphism group of a shift of subquadratic growth, Proc. Amer. Math. Soc., 144 (2016), 613-621. doi: 10.1090/proc12719.  Google Scholar

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V. Cyr and B. Kra, The automorphism group of a shift of linear growth: beyond transitivity, Forum Math. Sigma, 3 (2015), e5, 27pp. doi: 10.1017/fms.2015.3.  Google Scholar

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P. de la Harpe, Topics in Geometric Group Theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000.  Google Scholar

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S. Donoso, F. Durand, A. Maass and S. Petite, On automorphism groups of low complexity subshifts, Ergodic Theory Dynam. Systems, 36 (2016), 64-95. doi: 10.1017/etds.2015.70.  Google Scholar

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S. Donoso, F. Durand, A. Maass and S. Petite, Private, communication., ().   Google Scholar

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M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73.  Google Scholar

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Y. Guivarc'h, Groupes de Lie á croissance polynomiale, C. R. Acad. Sci. Paris Sér. A-B, 272 (1971), A1695-A1696.  Google Scholar

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G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory, 3 (1969), 320-375. doi: 10.1007/BF01691062.  Google Scholar

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M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42. doi: 10.2307/2371431.  Google Scholar

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K. H. Kim and F. W. Roush, On the automorphism groups of subshifts, Pure Math. Appl. Ser. B, 1 (1990), 203-230.  Google Scholar

[14]

V. Salo, Toeplitz subshift whose automorphism group is not finitely generated, Colloquium Mathematicum, (2016). doi: 10.4064/cm6463-2-2016.  Google Scholar

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V. Salo and I. Törmä, Block maps between primitive uniform and Pisot substitutions, Ergodic Theory and Dynam. Systems, 35 (2015), 2292-2310. doi: 10.1017/etds.2014.29.  Google Scholar

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L. van den Dries and A. Wilkie, Gromov's theorem on groups of polynomial growth and elementary logic, J. Algebra, 89 (1984), 349-374. doi: 10.1016/0021-8693(84)90223-0.  Google Scholar

show all references

References:
[1]

H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3), 25 (1972), 603-614.  Google Scholar

[2]

M. Boyle, D. Lind and D. Rudolph, The automorphism group of a shift of finite type, Trans. Amer. Math. Soc., 306 (1988), 71-114. doi: 10.1090/S0002-9947-1988-0927684-2.  Google Scholar

[3]

E. Coven, A. Quas and R. Yassawi, Computing automorphism groups of shifts, using atypical equivalence classes, Discrete Anal., (2016), 1-28. doi: 10.19086/da.611.  Google Scholar

[4]

V. Cyr and B. Kra, The automorphism group of a shift of subquadratic growth, Proc. Amer. Math. Soc., 144 (2016), 613-621. doi: 10.1090/proc12719.  Google Scholar

[5]

V. Cyr and B. Kra, The automorphism group of a shift of linear growth: beyond transitivity, Forum Math. Sigma, 3 (2015), e5, 27pp. doi: 10.1017/fms.2015.3.  Google Scholar

[6]

P. de la Harpe, Topics in Geometric Group Theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000.  Google Scholar

[7]

S. Donoso, F. Durand, A. Maass and S. Petite, On automorphism groups of low complexity subshifts, Ergodic Theory Dynam. Systems, 36 (2016), 64-95. doi: 10.1017/etds.2015.70.  Google Scholar

[8]

S. Donoso, F. Durand, A. Maass and S. Petite, Private, communication., ().   Google Scholar

[9]

M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73.  Google Scholar

[10]

Y. Guivarc'h, Groupes de Lie á croissance polynomiale, C. R. Acad. Sci. Paris Sér. A-B, 272 (1971), A1695-A1696.  Google Scholar

[11]

G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory, 3 (1969), 320-375. doi: 10.1007/BF01691062.  Google Scholar

[12]

M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42. doi: 10.2307/2371431.  Google Scholar

[13]

K. H. Kim and F. W. Roush, On the automorphism groups of subshifts, Pure Math. Appl. Ser. B, 1 (1990), 203-230.  Google Scholar

[14]

V. Salo, Toeplitz subshift whose automorphism group is not finitely generated, Colloquium Mathematicum, (2016). doi: 10.4064/cm6463-2-2016.  Google Scholar

[15]

V. Salo and I. Törmä, Block maps between primitive uniform and Pisot substitutions, Ergodic Theory and Dynam. Systems, 35 (2015), 2292-2310. doi: 10.1017/etds.2014.29.  Google Scholar

[16]

L. van den Dries and A. Wilkie, Gromov's theorem on groups of polynomial growth and elementary logic, J. Algebra, 89 (1984), 349-374. doi: 10.1016/0021-8693(84)90223-0.  Google Scholar

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