February  2018, 38(2): 835-866. doi: 10.3934/dcds.2018036

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

Received  January 2017 Revised  September 2017 Published  February 2018

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.

Citation: Michael Baake, John A. G. Roberts, Reem Yassawi. Reversing and extended symmetries of shift spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 835-866. doi: 10.3934/dcds.2018036
References:
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J. -P. Allouche and J. Shallit, Automatic Sequences Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511546563.

[2]

L. Arenas-CarmonaD. 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.

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M. Baake, Structure and representations of the hyperoctahedral group, J. Math. Phys., 25 (1984), 3171-3182. doi: 10.1063/1.526087.

[5]

M. BaakeF. 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. BaakeJ. 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. BaakeR. 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.

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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.

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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.

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E. M. Coven, Endomorphisms of substitution minimal sets, Z. Wahrscheinlichkeitsth. verw. Geb., 20 (1971/1972), 129-133. doi: 10.1007/BF00536290.

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E. M. Coven and G. A. Hedlund, Sequences with minimal block growth, Math. Systems Theory, 7 (1973), 138-153. doi: 10.1007/BF01762232.

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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.

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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.

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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.

[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. DonosoF. DurandA. 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 AbdalaouiM. 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. GiordanoI. 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. GoodsonA. del JuncoM. 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. HofO. 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. KimJ. 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. LeeK. 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-CarmonaD. 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. BaakeF. 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. BaakeJ. 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. BaakeR. 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. DonosoF. DurandA. 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 AbdalaouiM. 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. GiordanoI. 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. GoodsonA. del JuncoM. 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. HofO. 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. KimJ. 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. LeeK. 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.

Figure 1.  The chair inflation rule (upper left panel; rotated tiles are inflated to rotated patches), a legal patch with full $D_{4}$ symmetry (lower left) and a level-$3$ inflation patch generated from this legal seed (shaded; right panel). Note that this patch still has the full $D_{4}$ point symmetry (with respect to its centre), as will the infinite inflation tiling fixed point emerging from it
Figure 2.  The chair tiling seed of Figure 1, now turned into a patch of its symbolic representation via the recoding of Eq. (15). The relation between the purely geometric point symmetries in the tiling picture and the corresponding combinations of point symmetries and LEMs can be seen from this seed
Figure 3.  Illustration of the central configurational patch for Ledrappier's shift condition, which explains the relevance of the triangular lattice. Eq. (16) must be satisfied for the three vertices of all elementary $L$-triangles (shaded). The overall pattern of these triangles is preserved by all (extended) symmetries. The group $D^{}_{3}$ from Theorem 7 can now be viewed as the colour-preserving symmetry group of the 'distorted' hexagon as indicated around the origin
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