The automorphism group of a minimal shift of stretched exponential growth

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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

Abstract
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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.
[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.
[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.
[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.
[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.
[6]	P. de la Harpe, Topics in Geometric Group Theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000.
[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.
[8]	S. Donoso, F. Durand, A. Maass and S. Petite, Private communication.
[9]	M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73.
[10]	Y. Guivarc'h, Groupes de Lie á croissance polynomiale, C. R. Acad. Sci. Paris Sér. A-B, 272 (1971), A1695-A1696.
[11]	G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory, 3 (1969), 320-375. doi: 10.1007/BF01691062.
[12]	M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42. doi: 10.2307/2371431.
[13]	K. H. Kim and F. W. Roush, On the automorphism groups of subshifts, Pure Math. Appl. Ser. B, 1 (1990), 203-230.
[14]	V. Salo, Toeplitz subshift whose automorphism group is not finitely generated, Colloquium Mathematicum, (2016). doi: 10.4064/cm6463-2-2016.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[6]	P. de la Harpe, Topics in Geometric Group Theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000.
[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.
[8]	S. Donoso, F. Durand, A. Maass and S. Petite, Private communication.
[9]	M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73.
[10]	Y. Guivarc'h, Groupes de Lie á croissance polynomiale, C. R. Acad. Sci. Paris Sér. A-B, 272 (1971), A1695-A1696.
[11]	G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory, 3 (1969), 320-375. doi: 10.1007/BF01691062.
[12]	M. Morse and G. A. Hedlund, Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42. doi: 10.2307/2371431.
[13]	K. H. Kim and F. W. Roush, On the automorphism groups of subshifts, Pure Math. Appl. Ser. B, 1 (1990), 203-230.
[14]	V. Salo, Toeplitz subshift whose automorphism group is not finitely generated, Colloquium Mathematicum, (2016). doi: 10.4064/cm6463-2-2016.
[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.
[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.

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