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Reversing and extended symmetries of shift spaces
1. | Faculty of Mathematics, Universität Bielefeld, Box 100131,33501 Bielefeld, Germany |
2. | School of Mathematics and Statistics, UNSW, Sydney, NSW 2052, Australia |
3. | IRIF, Université Paris-Diderot — Paris 7, Case 7014,75205 Paris Cedex 13, France |
The reversing symmetry group is considered in the setting of symbolic dynamics. While this group is generally too big to be analysed in detail, there are interesting cases with some form of rigidity where one can determine all symmetries and reversing symmetries explicitly. They include Sturmian shifts as well as classic examples such as the Thue–Morse system with various generalisations or the Rudin–Shapiro system. We also look at generalisations of the reversing symmetry group to higher-dimensional shift spaces, then called the group of extended symmetries. We develop their basic theory for faithful $\mathbb{Z}^{d}$-actions, and determine the extended symmetry group of the chair tiling shift, which can be described as a model set, and of Ledrappier's shift, which is an example of algebraic origin.
References:
[1] |
J. -P. Allouche and J. Shallit,
Automatic Sequences Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511546563. |
[2] |
L. Arenas-Carmona, D. Berend and V. Bergelson,
Ledrappier's system is almost mixing of all orders, Ergodic Th. & Dynam. Syst., 28 (2008), 339-365.
doi: 10.1017/S0143385707000727. |
[3] |
J. Auslander,
Endomorphisms of minimal sets, Duke Math. J., 30 (1963), 605-614.
doi: 10.1215/S0012-7094-63-03065-5. |
[4] |
M. Baake,
Structure and representations of the hyperoctahedral group, J. Math. Phys., 25 (1984), 3171-3182.
doi: 10.1063/1.526087. |
[5] |
M. Baake, F. Gähler and U. Grimm,
Spectral and topological properties of a
family of generalised Thue–Morse sequences, J. Math. Phys., 53 (2012), 032701, 24pp.
doi: 10.1063/1.3688337. |
[6] |
M. Baake and U. Grimm,
Aperiodic Order. Vol. 1: A Mathematical Invitation Cambridge Univ. Press, Cambridge, 2013.
doi: 10.1017/CBO9781139025256. |
[7] |
M. Baake, J. Hermisson and P. A. B. Pleasants,
The torus parametrization of quasiperiodic LI classes, J. Phys. A: Math. Gen., 30 (1997), 3029-3056.
doi: 10.1088/0305-4470/30/9/016. |
[8] |
M. Baake, R. V. Moody and P. A. B. Pleasants,
Diffraction from visible lattice points
and $k$-th power free integers, Discr. Math., 221 (2000), 3-42.
doi: 10.1016/S0012-365X(99)00384-2. |
[9] |
M. Baake and J. A. G. Roberts,
Symmetries and reversing symmetries of
polynomial automorphisms of the plane, Nonlinearity, 18 (2005), 791-816.
doi: 10.1088/0951-7715/18/2/017. |
[10] |
M. Baake and J. A. G. Roberts,
Symmetries and reversing symmetries of toral automorphisms, Nonlinearity, 14 (2001), R1-R24.
doi: 10.1088/0951-7715/14/4/201. |
[11] |
M. Baake and J. A. G. Roberts. The structure of reversing symmetry groups,
The structure of reversing symmetry groups, Bull. Austral. Math. Soc., 73 (2006), 445-459.
doi: 10.1017/S0004972700035450. |
[12] |
M. Baake and T. Ward,
Planar dynamical systems with pure Lebesgue
diffraction spectrum, J. Stat. Phys., 140 (2010), 90-102.
doi: 10.1007/s10955-010-9984-x. |
[13] |
J. Berstel, L. Boasson, O. Carton and I. Fagnot, Infinite words without palindrome, preprint, arXiv: 0903.2382. |
[14] |
S. Bhattacharya and K. Schmidt,
Homoclinic points and isomorphism rigidity of algebraic $\mathbb{Z}^{d}$-actions on zero-dimensional compact Abelian groups, Israel J. Math., 137 (2003), 189-209.
doi: 10.1007/BF02785962. |
[15] |
S. Bhattacharya and T. Ward,
Finite entropy characterizes topological
rigidity on connected groups, Ergodic Th. & Dynam. Syst., 25 (2005), 365-373.
doi: 10.1017/S0143385704000501. |
[16] |
W. Bulatek and J. Kwiatkowski,
Strictly ergodic Toeplitz flows with positive
entropy and trivial centralizers, Studia Math., 103 (1992), 133-142.
doi: 10.4064/sm-103-2-133-142. |
[17] |
F. Cellarosi and Ya. G. Sinai,
Ergodic properties of square-free numbers, J. Europ. Math. Soc., 15 (2013), 1343-1374.
doi: 10.4171/JEMS/394. |
[18] |
E. M. Coven,
Endomorphisms of substitution minimal sets, Z. Wahrscheinlichkeitsth. verw.
Geb., 20 (1971/1972), 129-133.
doi: 10.1007/BF00536290. |
[19] |
E. M. Coven and G. A. Hedlund,
Sequences with minimal block growth, Math. Systems Theory, 7 (1973), 138-153.
doi: 10.1007/BF01762232. |
[20] |
E. M. Coven, A. Quas and R. Yassawi, Computing automorphism groups of shifts
using atypical equivalence classes Discrete Analysis 2016 (2016), 28pp.
doi: 10.19086/da.611. |
[21] |
V. Cyr and B. Kra,
The automorphism group of a shift of subquadratic growth, Proc. Amer. Math. Soc., 2 (2016), 613-621.
doi: 10.1090/proc12719. |
[22] |
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. |
[23] |
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. |
[24] |
M. Dekking,
The spectrum of dynamical systems
arising from substitutions of constant length, Z. Wahrscheinlichkeitsth. verw. Geb., 41 (1978), 221-239.
doi: 10.1007/BF00534241. |
[25] |
S. Donoso, F. Durand, A. Maass and S. Petite,
On automorphism groups of low complexity shifts, Ergodic Th. & Dynam. Syst., 36 (2016), 64-95.
doi: 10.1017/etds.2015.70. |
[26] |
X. Droubay and G. Pirillo,
Palindromes and Sturmian words, Theor. Comput. Sci., 223 (1999), 73-85.
doi: 10.1016/S0304-3975(97)00188-6. |
[27] |
F. Durand,
A characterization of substitutive
sequences using return words, Discr. Math., 179 (1998), 89-101.
doi: 10.1016/S0012-365X(97)00029-0. |
[28] |
M. Einsiedler and T. Ward,
Ergodic Theory with a View towards Number Theory Springer, London, 2011.
doi: 10.1007/978-0-85729-021-2. |
[29] |
E. H. El Abdalaoui, M. Lemańczyk and T. de la Rue,
A dynamical point of view on the
set of $\mathcal{B}$-free integers, Int. Math. Res. Notices, 2015 (2015), 7258-7286.
doi: 10.1093/imrn/rnu164. |
[30] |
T. Giordano, I. F. Putnam and C. Skau,
Full groups of Cantor minimal systems, Israel J. Math., 111 (1999), 285-320.
doi: 10.1007/BF02810689. |
[31] |
M. Golubitsky and I. Stewart, The Symmetry Perspective — From Equilibrium to Chaos in
Phase Space and Physical Space, Birkhäuser, Basel, 2002.
doi: 10.1007/978-3-0348-8167-8. |
[32] |
G. R. Goodson,
Inverse conjugacies and reversing symmetry groups, Amer. Math. Monthly, 106 (1999), 19-26.
doi: 10.2307/2589582. |
[33] |
G. Goodson, A. del Junco, M. Lemańczyk and D. Rudolph,
Ergodic transformation conjugate to
their inverses by involutions, Ergodic Th. & Dynam. Syst., 16 (1996), 97-124.
doi: 10.1017/S0143385700008737. |
[34] |
G. A. Hedlund,
Endomorphisms and automorphisms of
the shift dynamical systems, Math. Systems Th., 3 (1969), 320-375.
doi: 10.1007/BF01691062. |
[35] |
M. Hochman,
Genericity in topological dynamics, Ergodic Th. & Dynam. Syst., 28 (2008), 125-165.
doi: 10.1017/S0143385707000521. |
[36] |
A. Hof, O. Knill and B. Simon,
Singular continuous spectrum for
palindromic Schrödinger operators, Commun. Math. Phys., 174 (1995), 149-159.
doi: 10.1007/BF02099468. |
[37] |
K. Juschenko and N. Monod,
Cantor systems, piecewise translations
and simple amenable groups, Ann. of Math., 178 (2013), 775-787.
doi: 10.4007/annals.2013.178.2.7. |
[38] |
M. Keane,
Generalized Morse sequences, Z. Wahrscheinlichkeitsth. verw. Geb., 10 (1968), 335-353.
doi: 10.1007/BF00531855. |
[39] |
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. |
[40] |
B. P. Kitchens,
Symbolic Dynamics Springer, Berlin, 1998.
doi: 10.1007/978-3-642-58822-8. |
[41] |
B. Kitchens, Dynamics of Zd actions on Markov subgroups, in Topics in Symbolic Dynamics
and Applications, F. Blanchard, A. Maas and A. Nogueira (eds. ), Cambridge University Press,
Cambridge, (2000), pp. 89–122. |
[42] |
B. Kitchens and K. Schmidt,
Isomorphism rigidity of irreducible
algebraic $\mathbb{Z}^{d}$-actions, Invent. Math., 142 (2000), 559-577.
doi: 10.1007/PL00005793. |
[43] |
J. S. W. Lamb,
Reversing symmetries in dynamical systems, J. Phys. A: Math. Gen., 25 (1992), 925-937.
doi: 10.1088/0305-4470/25/4/028. |
[44] |
J. S. W. Lamb and J. A. G. Roberts,
Time-reversal symmetry in dynamical systems: A survey, Physica D, 112 (1998), 1-39.
doi: 10.1016/S0167-2789(97)00199-1. |
[45] |
F. Ledrappier,
Un champ markovien peut être d'entropie nulle et mélangeant, C. R. Acad.
Sci. Paris Sér. A-B, 287 (1978), A561-A563.
|
[46] |
J. Lee, K. K. Park and S. Shin,
Reversible topological Markov shifts, Ergodic Th. & Dynam. Syst., 26 (2006), 267-280.
doi: 10.1017/S0143385705000556. |
[47] |
D. A. Lind and B. Marcus,
An Introduction to Symbolic Dynamics and Coding Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511626302. |
[48] |
M. K. Mentzen,
Automorphisms of subshifts defined by
$\mathcal{B}$-free sets of integers, Coll. Math., 147 (2017), 87-94.
doi: 10.4064/cm6927-5-2016. |
[49] |
M. Morse and G. A. Hedlund,
Symbolic dynamics Ⅱ. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42.
doi: 10.2307/2371431. |
[50] |
A. G. O'Farrel and I. Short,
Reversibility in Dynamics and Group Theory Cambridge University Press, Cambridge, 2015.
doi: 10.1017/CBO9781139998321. |
[51] |
J. Olli,
Endomorphisms of Sturmian systems and
the discrete chair substitution tiling system, Discr. Cont. Dynam. Syst. A, 33 (2013), 4173-4186.
doi: 10.3934/dcds.2013.33.4173. |
[52] |
K. Petersen,
Ergodic Theory Cambridge University Press, Cambridge, 1983.
doi: 10.1017/CBO9780511608728. |
[53] |
M. Queffélec, Substitution Dynamical Systems — Spectral Analysis, LNM 1294, 2nd ed.,
Springer, Berlin, 2010. |
[54] |
J. A. G. Roberts and M. Baake,
Trace maps as 3D reversible dynamical
systems with an invariant, J. Stat. Phys., 74 (1994), 829-888.
doi: 10.1007/BF02188581. |
[55] |
E. A. Robinson,
On the table and the chair, Indag. Math., 10 (1999), 581-599.
doi: 10.1016/S0019-3577(00)87911-2. |
[56] |
K. Schmidt,
Dynamical Systems of Algebraic Origin Birkhäuser, Basel, 1995. |
[57] |
R. L. E. Schwarzenberger,
$N$-dimensional Crystallography Pitman, San Francisco, 1980. |
[58] |
M. B. Sevryuk,
Reversible Systems LNM 1211, Springer, Berlin, 1986.
doi: 10.1007/BFb0075877. |
[59] |
B. Tan,
Mirror substitutions and palindromic sequences, Theor. Comput. Sci., 389 (2007), 118-124.
doi: 10.1016/j.tcs.2007.08.003. |
[60] |
Ya. Vorobets,
On a substitution shift related to the Grigorchuk group, Proc. Steklov Inst. Math., 271 (2010), 306-321.
doi: 10.1134/S0081543810040218. |
show all references
References:
[1] |
J. -P. Allouche and J. Shallit,
Automatic Sequences Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511546563. |
[2] |
L. Arenas-Carmona, D. Berend and V. Bergelson,
Ledrappier's system is almost mixing of all orders, Ergodic Th. & Dynam. Syst., 28 (2008), 339-365.
doi: 10.1017/S0143385707000727. |
[3] |
J. Auslander,
Endomorphisms of minimal sets, Duke Math. J., 30 (1963), 605-614.
doi: 10.1215/S0012-7094-63-03065-5. |
[4] |
M. Baake,
Structure and representations of the hyperoctahedral group, J. Math. Phys., 25 (1984), 3171-3182.
doi: 10.1063/1.526087. |
[5] |
M. Baake, F. Gähler and U. Grimm,
Spectral and topological properties of a
family of generalised Thue–Morse sequences, J. Math. Phys., 53 (2012), 032701, 24pp.
doi: 10.1063/1.3688337. |
[6] |
M. Baake and U. Grimm,
Aperiodic Order. Vol. 1: A Mathematical Invitation Cambridge Univ. Press, Cambridge, 2013.
doi: 10.1017/CBO9781139025256. |
[7] |
M. Baake, J. Hermisson and P. A. B. Pleasants,
The torus parametrization of quasiperiodic LI classes, J. Phys. A: Math. Gen., 30 (1997), 3029-3056.
doi: 10.1088/0305-4470/30/9/016. |
[8] |
M. Baake, R. V. Moody and P. A. B. Pleasants,
Diffraction from visible lattice points
and $k$-th power free integers, Discr. Math., 221 (2000), 3-42.
doi: 10.1016/S0012-365X(99)00384-2. |
[9] |
M. Baake and J. A. G. Roberts,
Symmetries and reversing symmetries of
polynomial automorphisms of the plane, Nonlinearity, 18 (2005), 791-816.
doi: 10.1088/0951-7715/18/2/017. |
[10] |
M. Baake and J. A. G. Roberts,
Symmetries and reversing symmetries of toral automorphisms, Nonlinearity, 14 (2001), R1-R24.
doi: 10.1088/0951-7715/14/4/201. |
[11] |
M. Baake and J. A. G. Roberts. The structure of reversing symmetry groups,
The structure of reversing symmetry groups, Bull. Austral. Math. Soc., 73 (2006), 445-459.
doi: 10.1017/S0004972700035450. |
[12] |
M. Baake and T. Ward,
Planar dynamical systems with pure Lebesgue
diffraction spectrum, J. Stat. Phys., 140 (2010), 90-102.
doi: 10.1007/s10955-010-9984-x. |
[13] |
J. Berstel, L. Boasson, O. Carton and I. Fagnot, Infinite words without palindrome, preprint, arXiv: 0903.2382. |
[14] |
S. Bhattacharya and K. Schmidt,
Homoclinic points and isomorphism rigidity of algebraic $\mathbb{Z}^{d}$-actions on zero-dimensional compact Abelian groups, Israel J. Math., 137 (2003), 189-209.
doi: 10.1007/BF02785962. |
[15] |
S. Bhattacharya and T. Ward,
Finite entropy characterizes topological
rigidity on connected groups, Ergodic Th. & Dynam. Syst., 25 (2005), 365-373.
doi: 10.1017/S0143385704000501. |
[16] |
W. Bulatek and J. Kwiatkowski,
Strictly ergodic Toeplitz flows with positive
entropy and trivial centralizers, Studia Math., 103 (1992), 133-142.
doi: 10.4064/sm-103-2-133-142. |
[17] |
F. Cellarosi and Ya. G. Sinai,
Ergodic properties of square-free numbers, J. Europ. Math. Soc., 15 (2013), 1343-1374.
doi: 10.4171/JEMS/394. |
[18] |
E. M. Coven,
Endomorphisms of substitution minimal sets, Z. Wahrscheinlichkeitsth. verw.
Geb., 20 (1971/1972), 129-133.
doi: 10.1007/BF00536290. |
[19] |
E. M. Coven and G. A. Hedlund,
Sequences with minimal block growth, Math. Systems Theory, 7 (1973), 138-153.
doi: 10.1007/BF01762232. |
[20] |
E. M. Coven, A. Quas and R. Yassawi, Computing automorphism groups of shifts
using atypical equivalence classes Discrete Analysis 2016 (2016), 28pp.
doi: 10.19086/da.611. |
[21] |
V. Cyr and B. Kra,
The automorphism group of a shift of subquadratic growth, Proc. Amer. Math. Soc., 2 (2016), 613-621.
doi: 10.1090/proc12719. |
[22] |
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. |
[23] |
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. |
[24] |
M. Dekking,
The spectrum of dynamical systems
arising from substitutions of constant length, Z. Wahrscheinlichkeitsth. verw. Geb., 41 (1978), 221-239.
doi: 10.1007/BF00534241. |
[25] |
S. Donoso, F. Durand, A. Maass and S. Petite,
On automorphism groups of low complexity shifts, Ergodic Th. & Dynam. Syst., 36 (2016), 64-95.
doi: 10.1017/etds.2015.70. |
[26] |
X. Droubay and G. Pirillo,
Palindromes and Sturmian words, Theor. Comput. Sci., 223 (1999), 73-85.
doi: 10.1016/S0304-3975(97)00188-6. |
[27] |
F. Durand,
A characterization of substitutive
sequences using return words, Discr. Math., 179 (1998), 89-101.
doi: 10.1016/S0012-365X(97)00029-0. |
[28] |
M. Einsiedler and T. Ward,
Ergodic Theory with a View towards Number Theory Springer, London, 2011.
doi: 10.1007/978-0-85729-021-2. |
[29] |
E. H. El Abdalaoui, M. Lemańczyk and T. de la Rue,
A dynamical point of view on the
set of $\mathcal{B}$-free integers, Int. Math. Res. Notices, 2015 (2015), 7258-7286.
doi: 10.1093/imrn/rnu164. |
[30] |
T. Giordano, I. F. Putnam and C. Skau,
Full groups of Cantor minimal systems, Israel J. Math., 111 (1999), 285-320.
doi: 10.1007/BF02810689. |
[31] |
M. Golubitsky and I. Stewart, The Symmetry Perspective — From Equilibrium to Chaos in
Phase Space and Physical Space, Birkhäuser, Basel, 2002.
doi: 10.1007/978-3-0348-8167-8. |
[32] |
G. R. Goodson,
Inverse conjugacies and reversing symmetry groups, Amer. Math. Monthly, 106 (1999), 19-26.
doi: 10.2307/2589582. |
[33] |
G. Goodson, A. del Junco, M. Lemańczyk and D. Rudolph,
Ergodic transformation conjugate to
their inverses by involutions, Ergodic Th. & Dynam. Syst., 16 (1996), 97-124.
doi: 10.1017/S0143385700008737. |
[34] |
G. A. Hedlund,
Endomorphisms and automorphisms of
the shift dynamical systems, Math. Systems Th., 3 (1969), 320-375.
doi: 10.1007/BF01691062. |
[35] |
M. Hochman,
Genericity in topological dynamics, Ergodic Th. & Dynam. Syst., 28 (2008), 125-165.
doi: 10.1017/S0143385707000521. |
[36] |
A. Hof, O. Knill and B. Simon,
Singular continuous spectrum for
palindromic Schrödinger operators, Commun. Math. Phys., 174 (1995), 149-159.
doi: 10.1007/BF02099468. |
[37] |
K. Juschenko and N. Monod,
Cantor systems, piecewise translations
and simple amenable groups, Ann. of Math., 178 (2013), 775-787.
doi: 10.4007/annals.2013.178.2.7. |
[38] |
M. Keane,
Generalized Morse sequences, Z. Wahrscheinlichkeitsth. verw. Geb., 10 (1968), 335-353.
doi: 10.1007/BF00531855. |
[39] |
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. |
[40] |
B. P. Kitchens,
Symbolic Dynamics Springer, Berlin, 1998.
doi: 10.1007/978-3-642-58822-8. |
[41] |
B. Kitchens, Dynamics of Zd actions on Markov subgroups, in Topics in Symbolic Dynamics
and Applications, F. Blanchard, A. Maas and A. Nogueira (eds. ), Cambridge University Press,
Cambridge, (2000), pp. 89–122. |
[42] |
B. Kitchens and K. Schmidt,
Isomorphism rigidity of irreducible
algebraic $\mathbb{Z}^{d}$-actions, Invent. Math., 142 (2000), 559-577.
doi: 10.1007/PL00005793. |
[43] |
J. S. W. Lamb,
Reversing symmetries in dynamical systems, J. Phys. A: Math. Gen., 25 (1992), 925-937.
doi: 10.1088/0305-4470/25/4/028. |
[44] |
J. S. W. Lamb and J. A. G. Roberts,
Time-reversal symmetry in dynamical systems: A survey, Physica D, 112 (1998), 1-39.
doi: 10.1016/S0167-2789(97)00199-1. |
[45] |
F. Ledrappier,
Un champ markovien peut être d'entropie nulle et mélangeant, C. R. Acad.
Sci. Paris Sér. A-B, 287 (1978), A561-A563.
|
[46] |
J. Lee, K. K. Park and S. Shin,
Reversible topological Markov shifts, Ergodic Th. & Dynam. Syst., 26 (2006), 267-280.
doi: 10.1017/S0143385705000556. |
[47] |
D. A. Lind and B. Marcus,
An Introduction to Symbolic Dynamics and Coding Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511626302. |
[48] |
M. K. Mentzen,
Automorphisms of subshifts defined by
$\mathcal{B}$-free sets of integers, Coll. Math., 147 (2017), 87-94.
doi: 10.4064/cm6927-5-2016. |
[49] |
M. Morse and G. A. Hedlund,
Symbolic dynamics Ⅱ. Sturmian trajectories, Amer. J. Math., 62 (1940), 1-42.
doi: 10.2307/2371431. |
[50] |
A. G. O'Farrel and I. Short,
Reversibility in Dynamics and Group Theory Cambridge University Press, Cambridge, 2015.
doi: 10.1017/CBO9781139998321. |
[51] |
J. Olli,
Endomorphisms of Sturmian systems and
the discrete chair substitution tiling system, Discr. Cont. Dynam. Syst. A, 33 (2013), 4173-4186.
doi: 10.3934/dcds.2013.33.4173. |
[52] |
K. Petersen,
Ergodic Theory Cambridge University Press, Cambridge, 1983.
doi: 10.1017/CBO9780511608728. |
[53] |
M. Queffélec, Substitution Dynamical Systems — Spectral Analysis, LNM 1294, 2nd ed.,
Springer, Berlin, 2010. |
[54] |
J. A. G. Roberts and M. Baake,
Trace maps as 3D reversible dynamical
systems with an invariant, J. Stat. Phys., 74 (1994), 829-888.
doi: 10.1007/BF02188581. |
[55] |
E. A. Robinson,
On the table and the chair, Indag. Math., 10 (1999), 581-599.
doi: 10.1016/S0019-3577(00)87911-2. |
[56] |
K. Schmidt,
Dynamical Systems of Algebraic Origin Birkhäuser, Basel, 1995. |
[57] |
R. L. E. Schwarzenberger,
$N$-dimensional Crystallography Pitman, San Francisco, 1980. |
[58] |
M. B. Sevryuk,
Reversible Systems LNM 1211, Springer, Berlin, 1986.
doi: 10.1007/BFb0075877. |
[59] |
B. Tan,
Mirror substitutions and palindromic sequences, Theor. Comput. Sci., 389 (2007), 118-124.
doi: 10.1016/j.tcs.2007.08.003. |
[60] |
Ya. Vorobets,
On a substitution shift related to the Grigorchuk group, Proc. Steklov Inst. Math., 271 (2010), 306-321.
doi: 10.1134/S0081543810040218. |



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George Osipenko, Stephen Campbell. Applied symbolic dynamics: attractors and filtrations. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 43-60. doi: 10.3934/dcds.1999.5.43 |
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Michael Hochman. A note on universality in multidimensional symbolic dynamics. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 301-314. doi: 10.3934/dcdss.2009.2.301 |
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