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The mapping class group of a shift of finite type
1. | Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA |
2. | Algebra and Applications Research Unit, Department of Mathematics and Statistics, Prince of Songkla University, Songkhla, Thailand 90110 |
Let $(X_A,σ_{A})$ be a nontrivial irreducible shift of finite type (SFT), with $\mathscr{M}_A$ denoting its mapping class group: the group of flow equivalences of its mapping torus $\mathsf{S} X_A$, (i.e., self homeomorphisms of $\mathsf{S} X_A$ which respect the direction of the suspension flow) modulo the subgroup of flow equivalences of $\mathsf{S} X_A$ isotopic to the identity. We develop and apply machinery (flow codes, cohomology constraints) and provide context for the study of $\mathscr M_A$, and prove results including the following. $\mathscr{M}_A$ acts faithfully and $n$-transitively (for every $n$ in $\mathbb{N}$) by permutations on the set of circles of $\mathsf{S} X_A$. The center of $\mathscr{M}_A$ is trivial. The outer automorphism group of $\mathscr{M}_A$ is nontrivial. In many cases, $\text{Aut}(σ_{A})$ admits a nonspatial automorphism. For every SFT $(X_B,σ_B)$ flow equivalent to $(X_A,σ_{A})$, $\mathscr{M}_A$ contains embedded copies of ${\rm{Aut}}({\sigma _B})/\left\langle {{\sigma _B}} \right\rangle $, induced by return maps to invariant cross sections; but, elements of $\mathscr M_A$ not arising from flow equivalences with invariant cross sections are abundant. $\mathscr{M}_A$ is countable and has solvable word problem. $\mathscr{M}_A$ is not residually finite. Conjugacy classes of many (possibly all) involutions in $\mathscr M_A$ can be classified by the $G$-flow equivalence classes of associated $G$-SFTs, for $G = \mathbb{Z}/2\mathbb{Z}$. There are many open questions.
References:
[1] |
F. Blanchard and G. Hansel,
Systèmes codés, Theoret. Comput. Sci., 44 (1986), 17-49.
|
[2] |
R. Bowen and J. Franks,
Homology for zero-dimensional nonwandering sets, Ann. of Math. (2), 106 (1977), 73-92.
doi: 10.2307/1971159. |
[3] |
M. Boyle,
Flow equivalence of shifts of finite type via positive factorizations, Pacific J. Math., 204 (2002), 273-317.
doi: 10.2140/pjm.2002.204.273. |
[4] |
M. Boyle, Positive K-theory and symbolic dynamics, in Dynamics and Randomness (Santiago, 2000), Nonlinear Phenom. Complex Systems, 7, Kluwer Acad. Publ., 2002, 31-52. |
[5] |
M. Boyle, T. Carlsen and S. Eilers, Flow equivalence of G-SFTs, arXiv: 1512.05238, 2015. Google Scholar |
[6] |
M. Boyle, T. Carlsen and S. Eilers, Flow equivalence of sofic shifts, Israel J. Math., to appear; arXiv: 1511.03481, 2015. Google Scholar |
[7] |
M. Boyle, T. Carlsen and S. Eilers,
Flow equivalence and isotopy for subshifts, Dyn. Syst., 32 (2017), 305-325.
doi: 10.1080/14689367.2016.1207753. |
[8] |
M. Boyle and U.-R. Fiebig,
The action of inert finite-order automorphisms on finite subsystems of the shift, Ergodic Theory Dynam. Systems, 11 (1991), 413-425.
|
[9] |
M. Boyle and D. Handelman,
Orbit equivalence, flow equivalence and ordered cohomology, Israel J. Math., 95 (1996), 169-210.
doi: 10.1007/BF02761039. |
[10] |
M. Boyle and D. Huang,
Poset block equivalence of integral matrices, Trans. Amer. Math. Soc., 355 (2003), 3861-3886.
doi: 10.1090/S0002-9947-03-02947-7. |
[11] |
M. Boyle and W. Krieger,
Periodic points and automorphisms of the shift, Trans. Amer. Math. Soc., 302 (1987), 125-149.
doi: 10.1090/S0002-9947-1987-0887501-5. |
[12] |
M. Boyle and W. Krieger, Almost Markov and shift equivalent sofic systems, in Dynamical Systems (College Park, MD, 1986-87), Lecture Notes in Math., 1342, Springer, Berlin, 1988, 33-93. |
[13] |
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. |
[14] |
M. Boyle and S. Schmieding,
Finite group extensions of shifts of finite type: K-theory, Parry and Livšic, Ergodic Theory Dynam. Systems, 37 (2017), 1026-1059.
doi: 10.1017/etds.2015.87. |
[15] |
M. Boyle and M. C. Sullivan,
Equivariant flow equivalence for shifts of finite type, by matrix equivalence over group rings, Proc. London Math. Soc. (3), 91 (2005), 184-214.
doi: 10.1112/S0024611505015285. |
[16] |
M. Boyle and J. B. Wagoner, Positive algebraic K-theory and shifts of finite type, in Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge, 2004, 45-66. |
[17] |
V. Capraro and M. Lupini, Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture, Lecture Notes in Mathematics, 2136, Springer, Cham, 2015. With an appendix by Vladimir Pestov. |
[18] |
S. Chuysurichay, Positive Rational Strong Shift Equivalence and the Mapping Class Group of a Shift of Finite Type, Thesis (Ph.D.)-University of Maryland, College Park, 2011, 95 pp, ProQuest LLC, Ann Arbor, MI. |
[19] |
E. M. Coven, A. Quas and R. Yassawi, Computing automorphism groups of shifts using atypical equivalence classes, Discrete Anal., (2016), Paper No. 3, 28pp. |
[20] |
V. Cyr, J. Franks, B. Kra and S. Petite, Distortion and the automorphism group of a shift, J. Mod. Dyn., 13 (2018), 147–161.
doi: 10.3934/jmd.2018015. |
[21] |
V. Cyr and B. Kra,
The automorphism group of a minimal shift of stretched exponential growth, J. Mod. Dyn., 10 (2016), 483-495.
doi: 10.3934/jmd.2016.10.483. |
[22] |
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. |
[23] |
S. Eilers, G. Restorff, E. Ruiz and A. P. W. Sorensen,
The complete classification of unital graph C*-algebras: Geometric and strong, Canad. J. Math., 70 (2018), 294-353.
doi: 10.4153/CJM-2017-016-7. |
[24] |
B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012. |
[25] |
U.-R. Fiebig,
Periodic points and finite group actions on shifts of finite type, Ergodic Theory Dynam. Systems, 13 (1993), 485-514.
|
[26] |
R. J. Fokkink, The Structure of Trajectories, Thesis (Ph.D.)-Technische Universiteit Delft (The Netherlands), 1991, 112pp, ProQuest LLC, Ann Arbor, MI. |
[27] |
J. Franks,
Flow equivalence of subshifts of finite type, Ergodic Theory Dynam. Systems, 4 (1984), 53-66.
|
[28] |
T. Giordano, I. F. Putnam and C. F. Skau,
Topological orbit equivalence and $ C^*$-crossed products, J. Reine Angew. Math., 469 (1995), 51-111.
|
[29] |
T. Giordano, I. F. Putnam and C. F. Skau,
Full groups of Cantor minimal systems, Israel J. Math., 111 (1999), 285-320.
doi: 10.1007/BF02810689. |
[30] |
R. I. Grigorchuk and K. S. Medinets,
On the algebraic properties of topological full groups, Mat. Sb., 205 (2014), 87-108.
|
[31] |
G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory, 3 (1969), 320-375.
doi: 10.1007/BF01691062. |
[32] |
M. Hochman,
On the automorphism groups of multidimensional shifts of finite type, Ergodic Theory Dynam. Systems, 30 (2010), 809-840.
doi: 10.1017/S0143385709000248. |
[33] |
K. Juschenko and N. Monod,
Cantor systems, piecewise translations and simple amenable groups, Ann. of Math. (2), 178 (2013), 775-787.
doi: 10.4007/annals.2013.178.2.7. |
[34] |
K. H. Kim and F. W. Roush,
On the automorphism groups of subshifts, Pure Math. Appl. Ser. B, 1 (1990), 203-230 (1991).
|
[35] |
K. H. Kim and F. W. Roush,
Free $ Z_p$ actions on subshifts, Pure Math. Appl., 8 (1997), 293-322.
|
[36] |
K. H. Kim, F. W. Roush and J. B. Wagoner,
Automorphisms of the dimension group and gyration numbers, J. Amer. Math. Soc., 5 (1992), 191-212.
doi: 10.1090/S0894-0347-1992-1124983-3. |
[37] |
K. H. Kim, F. W. Roush and S. G. Williams, Duality and its consequences for ordered cohomology of finite type subshifts, in Combinatorial & Computational Mathematics (Pohang, 2000), World Sci. Publ., River Edge, NJ, 2001, 243-265. |
[38] |
Y.-O. Kim, J. Lee and K. K. Park,
A zeta function for flip systems, Pacific J. Math., 209 (2003), 289-301.
doi: 10.2140/pjm.2003.209.289. |
[39] |
D. A. Lind,
The entropies of topological Markov shifts and a related class of algebraic integers, Ergodic Theory Dynam. Systems, 4 (1984), 283-300.
|
[40] |
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. |
[41] |
N. Long,
Fixed point shifts of inert involutions, Discrete Contin. Dyn. Syst., 25 (2009), 1297-1317.
doi: 10.3934/dcds.2009.25.1297. |
[42] |
K. Matsumoto and H. Matui,
Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras, Kyoto J. Math., 54 (2014), 863-877.
doi: 10.1215/21562261-2801849. |
[43] |
H. Matui,
Topological full groups of one-sided shifts of finite type, J. Reine Angew. Math., 705 (2015), 35-84.
|
[44] |
M. Nasu, Topological conjugacy for sofic systems and extensions of automorphisms of finite subsystems of topological Markov shifts, in Dynamical Systems (College Park, MD, 1986-87), Lecture Notes in Math., 1342, Springer, Berlin, 1988, 564-607. |
[45] |
W. Parry and D. Sullivan,
A topological invariant of flows on 1-dimensional spaces, Topology, 14 (1975), 297-299.
doi: 10.1016/0040-9383(75)90012-9. |
[46] |
V. G. Pestov,
Hyperlinear and sofic groups: A brief guide, Bull. Symbolic Logic, 14 (2008), 449-480.
doi: 10.2178/bsl/1231081461. |
[47] |
G. Restorff,
Classification of Cuntz-Krieger algebras up to stable isomorphism, J. Reine Angew. Math., 598 (2006), 185-210.
|
[48] |
M. Rørdam,
Classification of Cuntz-Krieger algebras, K-Theory, 9 (1995), 31-58.
doi: 10.1007/BF00965458. |
[49] |
J. Patrick Ryan,
The shift and commutivity. Ⅱ, Math. Systems Theory, 8 (1974/75), 249-250.
doi: 10.1007/BF01762673. |
[50] |
V. Salo, Groups and monoids of cellular automata, in Cellular Automata and Discrete Complex Systems, Lecture Notes in Comput. Sci., 9099, Springer, Heidelberg, 2015, 17-45. |
[51] |
V. Salo and I. Törmä,
Block maps between primitive uniform and Pisot substitutions, Ergodic Theory Dynam. Systems, 35 (2015), 2292-2310.
doi: 10.1017/etds.2014.29. |
[52] |
M. Schraudner,
On the algebraic properties of the automorphism groups of countable-state Markov shifts, Ergodic Theory Dynam. Systems, 26 (2006), 551-583.
doi: 10.1017/S0143385705000507. |
[53] |
S. Schwartzman,
Asymptotic cycles, Ann. of Math. (2), 66 (1957), 270-284.
doi: 10.2307/1969999. |
[54] |
J. B. Wagoner,
Strong shift equivalence theory and the shift equivalence problem, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 271-296.
doi: 10.1090/S0273-0979-99-00798-3. |
[55] |
J. B. Wagoner, Strong shift equivalence and $ K_2$ of the dual numbers, J. Reine Angew. Math., 521 (2000), 119-160, with an appendix by K. H. Kim and F. W. Roush. |
[56] |
B. Weiss,
Sofic groups and dynamical systems, Ergodic Theory and Harmonic Analysis (Mumbai, 1999), Sankhyā Ser. A, 62 (2000), 350-359.
|
show all references
References:
[1] |
F. Blanchard and G. Hansel,
Systèmes codés, Theoret. Comput. Sci., 44 (1986), 17-49.
|
[2] |
R. Bowen and J. Franks,
Homology for zero-dimensional nonwandering sets, Ann. of Math. (2), 106 (1977), 73-92.
doi: 10.2307/1971159. |
[3] |
M. Boyle,
Flow equivalence of shifts of finite type via positive factorizations, Pacific J. Math., 204 (2002), 273-317.
doi: 10.2140/pjm.2002.204.273. |
[4] |
M. Boyle, Positive K-theory and symbolic dynamics, in Dynamics and Randomness (Santiago, 2000), Nonlinear Phenom. Complex Systems, 7, Kluwer Acad. Publ., 2002, 31-52. |
[5] |
M. Boyle, T. Carlsen and S. Eilers, Flow equivalence of G-SFTs, arXiv: 1512.05238, 2015. Google Scholar |
[6] |
M. Boyle, T. Carlsen and S. Eilers, Flow equivalence of sofic shifts, Israel J. Math., to appear; arXiv: 1511.03481, 2015. Google Scholar |
[7] |
M. Boyle, T. Carlsen and S. Eilers,
Flow equivalence and isotopy for subshifts, Dyn. Syst., 32 (2017), 305-325.
doi: 10.1080/14689367.2016.1207753. |
[8] |
M. Boyle and U.-R. Fiebig,
The action of inert finite-order automorphisms on finite subsystems of the shift, Ergodic Theory Dynam. Systems, 11 (1991), 413-425.
|
[9] |
M. Boyle and D. Handelman,
Orbit equivalence, flow equivalence and ordered cohomology, Israel J. Math., 95 (1996), 169-210.
doi: 10.1007/BF02761039. |
[10] |
M. Boyle and D. Huang,
Poset block equivalence of integral matrices, Trans. Amer. Math. Soc., 355 (2003), 3861-3886.
doi: 10.1090/S0002-9947-03-02947-7. |
[11] |
M. Boyle and W. Krieger,
Periodic points and automorphisms of the shift, Trans. Amer. Math. Soc., 302 (1987), 125-149.
doi: 10.1090/S0002-9947-1987-0887501-5. |
[12] |
M. Boyle and W. Krieger, Almost Markov and shift equivalent sofic systems, in Dynamical Systems (College Park, MD, 1986-87), Lecture Notes in Math., 1342, Springer, Berlin, 1988, 33-93. |
[13] |
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. |
[14] |
M. Boyle and S. Schmieding,
Finite group extensions of shifts of finite type: K-theory, Parry and Livšic, Ergodic Theory Dynam. Systems, 37 (2017), 1026-1059.
doi: 10.1017/etds.2015.87. |
[15] |
M. Boyle and M. C. Sullivan,
Equivariant flow equivalence for shifts of finite type, by matrix equivalence over group rings, Proc. London Math. Soc. (3), 91 (2005), 184-214.
doi: 10.1112/S0024611505015285. |
[16] |
M. Boyle and J. B. Wagoner, Positive algebraic K-theory and shifts of finite type, in Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge, 2004, 45-66. |
[17] |
V. Capraro and M. Lupini, Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture, Lecture Notes in Mathematics, 2136, Springer, Cham, 2015. With an appendix by Vladimir Pestov. |
[18] |
S. Chuysurichay, Positive Rational Strong Shift Equivalence and the Mapping Class Group of a Shift of Finite Type, Thesis (Ph.D.)-University of Maryland, College Park, 2011, 95 pp, ProQuest LLC, Ann Arbor, MI. |
[19] |
E. M. Coven, A. Quas and R. Yassawi, Computing automorphism groups of shifts using atypical equivalence classes, Discrete Anal., (2016), Paper No. 3, 28pp. |
[20] |
V. Cyr, J. Franks, B. Kra and S. Petite, Distortion and the automorphism group of a shift, J. Mod. Dyn., 13 (2018), 147–161.
doi: 10.3934/jmd.2018015. |
[21] |
V. Cyr and B. Kra,
The automorphism group of a minimal shift of stretched exponential growth, J. Mod. Dyn., 10 (2016), 483-495.
doi: 10.3934/jmd.2016.10.483. |
[22] |
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. |
[23] |
S. Eilers, G. Restorff, E. Ruiz and A. P. W. Sorensen,
The complete classification of unital graph C*-algebras: Geometric and strong, Canad. J. Math., 70 (2018), 294-353.
doi: 10.4153/CJM-2017-016-7. |
[24] |
B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012. |
[25] |
U.-R. Fiebig,
Periodic points and finite group actions on shifts of finite type, Ergodic Theory Dynam. Systems, 13 (1993), 485-514.
|
[26] |
R. J. Fokkink, The Structure of Trajectories, Thesis (Ph.D.)-Technische Universiteit Delft (The Netherlands), 1991, 112pp, ProQuest LLC, Ann Arbor, MI. |
[27] |
J. Franks,
Flow equivalence of subshifts of finite type, Ergodic Theory Dynam. Systems, 4 (1984), 53-66.
|
[28] |
T. Giordano, I. F. Putnam and C. F. Skau,
Topological orbit equivalence and $ C^*$-crossed products, J. Reine Angew. Math., 469 (1995), 51-111.
|
[29] |
T. Giordano, I. F. Putnam and C. F. Skau,
Full groups of Cantor minimal systems, Israel J. Math., 111 (1999), 285-320.
doi: 10.1007/BF02810689. |
[30] |
R. I. Grigorchuk and K. S. Medinets,
On the algebraic properties of topological full groups, Mat. Sb., 205 (2014), 87-108.
|
[31] |
G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory, 3 (1969), 320-375.
doi: 10.1007/BF01691062. |
[32] |
M. Hochman,
On the automorphism groups of multidimensional shifts of finite type, Ergodic Theory Dynam. Systems, 30 (2010), 809-840.
doi: 10.1017/S0143385709000248. |
[33] |
K. Juschenko and N. Monod,
Cantor systems, piecewise translations and simple amenable groups, Ann. of Math. (2), 178 (2013), 775-787.
doi: 10.4007/annals.2013.178.2.7. |
[34] |
K. H. Kim and F. W. Roush,
On the automorphism groups of subshifts, Pure Math. Appl. Ser. B, 1 (1990), 203-230 (1991).
|
[35] |
K. H. Kim and F. W. Roush,
Free $ Z_p$ actions on subshifts, Pure Math. Appl., 8 (1997), 293-322.
|
[36] |
K. H. Kim, F. W. Roush and J. B. Wagoner,
Automorphisms of the dimension group and gyration numbers, J. Amer. Math. Soc., 5 (1992), 191-212.
doi: 10.1090/S0894-0347-1992-1124983-3. |
[37] |
K. H. Kim, F. W. Roush and S. G. Williams, Duality and its consequences for ordered cohomology of finite type subshifts, in Combinatorial & Computational Mathematics (Pohang, 2000), World Sci. Publ., River Edge, NJ, 2001, 243-265. |
[38] |
Y.-O. Kim, J. Lee and K. K. Park,
A zeta function for flip systems, Pacific J. Math., 209 (2003), 289-301.
doi: 10.2140/pjm.2003.209.289. |
[39] |
D. A. Lind,
The entropies of topological Markov shifts and a related class of algebraic integers, Ergodic Theory Dynam. Systems, 4 (1984), 283-300.
|
[40] |
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995. |
[41] |
N. Long,
Fixed point shifts of inert involutions, Discrete Contin. Dyn. Syst., 25 (2009), 1297-1317.
doi: 10.3934/dcds.2009.25.1297. |
[42] |
K. Matsumoto and H. Matui,
Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras, Kyoto J. Math., 54 (2014), 863-877.
doi: 10.1215/21562261-2801849. |
[43] |
H. Matui,
Topological full groups of one-sided shifts of finite type, J. Reine Angew. Math., 705 (2015), 35-84.
|
[44] |
M. Nasu, Topological conjugacy for sofic systems and extensions of automorphisms of finite subsystems of topological Markov shifts, in Dynamical Systems (College Park, MD, 1986-87), Lecture Notes in Math., 1342, Springer, Berlin, 1988, 564-607. |
[45] |
W. Parry and D. Sullivan,
A topological invariant of flows on 1-dimensional spaces, Topology, 14 (1975), 297-299.
doi: 10.1016/0040-9383(75)90012-9. |
[46] |
V. G. Pestov,
Hyperlinear and sofic groups: A brief guide, Bull. Symbolic Logic, 14 (2008), 449-480.
doi: 10.2178/bsl/1231081461. |
[47] |
G. Restorff,
Classification of Cuntz-Krieger algebras up to stable isomorphism, J. Reine Angew. Math., 598 (2006), 185-210.
|
[48] |
M. Rørdam,
Classification of Cuntz-Krieger algebras, K-Theory, 9 (1995), 31-58.
doi: 10.1007/BF00965458. |
[49] |
J. Patrick Ryan,
The shift and commutivity. Ⅱ, Math. Systems Theory, 8 (1974/75), 249-250.
doi: 10.1007/BF01762673. |
[50] |
V. Salo, Groups and monoids of cellular automata, in Cellular Automata and Discrete Complex Systems, Lecture Notes in Comput. Sci., 9099, Springer, Heidelberg, 2015, 17-45. |
[51] |
V. Salo and I. Törmä,
Block maps between primitive uniform and Pisot substitutions, Ergodic Theory Dynam. Systems, 35 (2015), 2292-2310.
doi: 10.1017/etds.2014.29. |
[52] |
M. Schraudner,
On the algebraic properties of the automorphism groups of countable-state Markov shifts, Ergodic Theory Dynam. Systems, 26 (2006), 551-583.
doi: 10.1017/S0143385705000507. |
[53] |
S. Schwartzman,
Asymptotic cycles, Ann. of Math. (2), 66 (1957), 270-284.
doi: 10.2307/1969999. |
[54] |
J. B. Wagoner,
Strong shift equivalence theory and the shift equivalence problem, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 271-296.
doi: 10.1090/S0273-0979-99-00798-3. |
[55] |
J. B. Wagoner, Strong shift equivalence and $ K_2$ of the dual numbers, J. Reine Angew. Math., 521 (2000), 119-160, with an appendix by K. H. Kim and F. W. Roush. |
[56] |
B. Weiss,
Sofic groups and dynamical systems, Ergodic Theory and Harmonic Analysis (Mumbai, 1999), Sankhyā Ser. A, 62 (2000), 350-359.
|
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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 |
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