November  2022, 42(11): 5399-5435. doi: 10.3934/dcds.2022105

Spectral theory of spin substitutions

1. 

Department of Mathematics and Statistics, Vassar College, Box 248, Poughkeepsie, NY 12604, USA

2. 

Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

3. 

School of Mathematics and Statistics, Open University, Walton Hall, Kents Hill, Milton Keynes MK7 6AA, United Kingdom

*Corresponding author: Neil Mañibo

Dedicated to our late friend and colleague, Uwe Grimm
The second author is funded by the German Research Foundation (DFG, Deutsche Forschungs-gemeinschaft), via SFB 1283/2 2021-317210226

Received  August 2021 Revised  April 2022 Published  November 2022 Early access  August 2022

We introduce substitutions in $ {\mathbb{Z}}^m $ which have non-rectangular domains based on an endomorphism $ Q $ of $ {\mathbb{Z}}^m $ and a set $ {\mathcal D} $ of coset representatives of $ {\mathbb{Z}}^m/Q{\mathbb{Z}}^m $, which we call digit substitutions. Using a finite abelian 'spin' group we define 'spin digit substitutions' and their subshifts $ ({\Sigma}, {\mathbb{Z}}^m) $. Conditions under which the subshift is measure-theoretically isomorphic to a group extension of an $ m $-dimensional odometer are given, inducing a complete decomposition of the function space $ L^{2}({\Sigma},\mu) $. This enables the use of group characters in $ {\widehat{G}} $ to derive substitutive factors and analyze the spectra of specific subspaces. We provide general sufficient criteria for the existence of pure point, absolutely continuous, and singular continuous spectral measures, together with some bounds on their spectral multiplicity.

Citation: Natalie Priebe Frank, Neil Mañibo. Spectral theory of spin substitutions. Discrete and Continuous Dynamical Systems, 2022, 42 (11) : 5399-5435. doi: 10.3934/dcds.2022105
References:
[1]

I. Abou and P. Liardet, Flots chaînés, Proceedings of the Sixth Congress of Romanian Mathematicians Vol.1, L. Beznea, V. Brinzanescu, R. Purice, et.al. (eds.), Editura Academiei Române, Bucharest (2009), 401–432.

[2]

J.-P. Allouche and P. Liardet, Generalized Rudin–Shapiro sequences, Acta Arith., 60 (1991), 1-27.  doi: 10.4064/aa-60-1-1-27.

[3]

M. BaakeN. P. FrankU. Grimm and E. A. Robinson, Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction, Studia Math., 247 (2019), 109-154.  doi: 10.4064/sm170613-10-3.

[4]

M. Baake, F. Gähler and U. Grimm, Examples of substitution systems and their factors, J. Int. Seq., 16 (2013), 13.2.14, 18 pp.

[5]

M. BaakeF. Gähler and N. Mañibo, Renormalisation of pair correlation measures for primitive inflation rules and absence of absolutely continuous diffraction, Commun. Math. Phys., 370 (2019), 591-635.  doi: 10.1007/s00220-019-03500-w.

[6] M. Baake and U. Grimm, Aperiodic Order Volume 1. A Mathematical Invitation, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139025256.
[7]

M. Baake and U. Grimm, Squirals and beyond: Substitution tilings with singular continuous spectrum, Ergodic Th. & Dynam. Syst., 34 (2014), 1077-1102.  doi: 10.1017/etds.2012.191.

[8]

M. Baake and D. Lenz, Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra, Ergod. Th. & Dynam. Syst., 24 (2004), 1867-1893.  doi: 10.1017/S0143385704000318.

[9]

M. BaakeD. Lenz and A. van Enter, Dynamical versus diffraction spectrum for structures with finite local complexity, Ergod. Th. & Dynam. Syst., 35 (2015), 2017-2043.  doi: 10.1017/etds.2014.28.

[10]

E. Bannai and E. Bannai, Spin models on finite cyclic groups, J. Alg. Combin., 3 (1994), 243-259.  doi: 10.1023/A:1022407800541.

[11] L. Barreira and Y. Pesin, Nonuniform Hyperbolicity, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9781107326026.
[12]

A. Bartlett, Spectral theory of $\mathbb{Z}^d$ substitutions, Ergodic Th. & Dynam. Syst., 38 (2018), 1289-1341.  doi: 10.1017/etds.2016.66.

[13]

A. Berlinkov and B. Solomyak, Singular substitutions of constant length, Ergodic Th. & Dynam. Syst., 39 (2019), 2384-2402.  doi: 10.1017/etds.2017.133.

[14]

A. I. Bufetov and B. Solomyak, A spectral cocycle for substitution systems and translation flows, J. Anal. Math., 141 (2020), 165-205.  doi: 10.1007/s11854-020-0127-2.

[15]

C. Cabezas, Homomorphisms between multidimensional constant-shape substitutions, preprint, arXiv: 2106.10504.

[16]

L. ChanU. Grimm and I. Short, Substitution-based structures with absolutely continuous spectrum, Indag. Math., 29 (2018), 1072-1086.  doi: 10.1016/j.indag.2018.05.009.

[17]

M. I. Cortez, $ {\mathbb{Z}}^d$ Toeplitz arrays, Discr. Contin. Dynam. Syst. A, 15 (2006), 859-881.  doi: 10.3934/dcds.2006.15.859.

[18]

E. M. Coven and A. Meyerowitz, Tiling the integers with translates of one finite set, J. Algebra, 212 (1999), 161-174.  doi: 10.1006/jabr.1998.7628.

[19]

J. Dubédat, Topics on abelian spin models and related problems, Probab. Surveys, 8 (2011), 374-402.  doi: 10.1214/11-PS187.

[20]

E. H. el Abdalaoui and M. Lemańczyk, Approximately transitive dynamical systems and simple spectrum, Arch. Math., 97 (2011), 187-197.  doi: 10.1007/s00013-011-0285-7.

[21]

T. Fernique, Local rule substitutions and stepped surfaces, Theoret. Comp. Sci., 380 (2007), 317-329.  doi: 10.1016/j.tcs.2007.03.021.

[22]

N. P. Frank, Introduction to hierarchical tiling dynamical systems, In Substitution and Tiling Dynamics: Introduction to Self-inducing Structures, S. Akiyama and P. Arnoux (eds.), LNM 2773, Springer, Cham (2020), 33–95. doi: 10.1007/978-3-030-57666-0_2.

[23]

N. P. Frank, Multidimensional constant-length substitution sequences, Topology & Appl., 152 (2005), 44-69.  doi: 10.1016/j.topol.2004.08.014.

[24]

N. P. Frank, Substitution sequences in $\mathbb{Z}^d$ with a nonsimple Lebesgue component in the spectrum, Ergodic Th. & Dynam. Syst., 23 (2003), 519-532.  doi: 10.1017/S0143385702001256.

[25]

N. P. Frank and L. Sadun, Fusion: A general framework for hierarchical tilings of $\mathbb{R}^d$, Geom. Dedicata, 171 (2014), 149-186.  doi: 10.1007/s10711-013-9893-7.

[26]

N. P. Frank and L. Sadun, Fusion tilings with infinite local complexity, Top. Proc. 43 (2014) 235–276.

[27]

R. Greenfeld and T. Tao, The structure of translational tilings in $\mathbb{Z}^d$, Discr. Anal., 16 (2021), arXiv: 2010.03254, 28 pp. doi: 10.19086/da.

[28]

K. Gröchenig and A. Haas, Self-similar lattice tilings, J. Fourier Anal. Appl., 1 (1994), 131-170.  doi: 10.1007/s00041-001-4007-6.

[29]

P. R. Halmos and J. von Neumann, Operator methods in classical mechanics II, Ann. Math., 43 (1942), 332-350.  doi: 10.2307/1968872.

[30]

H. Helson, Cocycles on the circle, J. Oper. Theory, 16 (1986), 189-199. 

[31]

R. Kenyon, Self-replicating tilings, Contemp. Math., 135 (1992), 239-263.  doi: 10.1090/conm/135/1185093.

[32]

J. C. Lagarias and Y. Wang, Integral self-affine tiles in $\mathbb{R}^n$ I. standard and nonstandard digit sets, J. London Math. Soc., 54 (1996), 161-179.  doi: 10.1112/jlms/54.1.161.

[33]

J. C. Lagarias and Y. Wang, Self-affine tiles in $\mathbb{R}^n$, Adv. Math., 121 (1996), 21-49.  doi: 10.1006/aima.1996.0045.

[34]

J.-Y. LeeR. V. Moody and B. Solomyak, Consequences of pure point diffraction spectra for multiset substitution systems, Discrete Comput. Geom., 29 (2003), 525-560.  doi: 10.1007/s00454-003-0781-z.

[35]

J.-Y. LeeR. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra, Ann. Henri Poincaré, 3 (2002), 1003-1018.  doi: 10.1007/s00023-002-8646-1.

[36]

D. Lenz, Spectral theory of dynamical systems as diffraction theory of sampling functions, Monats. Math., 192 (2020), 625-649.  doi: 10.1007/s00605-020-01419-2.

[37]

N. Mañibo, Lyapunov exponents for binary substitutions of constant length, J. Math. Phys., 58 (2017), 113504, 9 pp. doi: 10.1063/1.4993169.

[38]

N. Mañibo, D. Rust and J. Walton, Spectral properties of substitutions on compact alphabets, preprint, arXiv: 2108.01762.

[39]

R. Meshulam, On subsets of finite abelian groups with no $3$-term arithmetic progression, J. Combin. Theor. A, 71 (1995), 168-172.  doi: 10.1016/0097-3165(95)90024-1.

[40]

M. G. Nadkarni, The skew product, In Spectral Theory of Dynamical Systems, R. B. Bapat, V. S. Borkar, P. Chaudhuri, et.al. (eds.), Hindustan Book Agency, Gurgaon (1998), 37–39.

[41]

M. Queffélec, Substitution Dynamical Systems–Spectral Analysis, 2nd. ed., LNM 1294, Springer, Berlin, 2010. doi: 10.1007/978-3-642-11212-6.

[42]

M. Queffélec, Une nouvelle properiété des suites de Rudin–Shapiro, Ann. Inst. Fourier, 37 (1987), 115-138.  doi: 10.5802/aif.1089.

[43]

E. A. Robinson, Non-abelian extensions have nonsimple spectrum, Compos. Math., 65 (1988), 155-170.

[44]

B. Solomyak, Dynamics of self-similar tilings, Ergodic Th. & Dynam. Syst., 17 (1997), 695–738 and Ergodic Th. & Dynam. Syst., 19 (1999), 1685 (erratum). doi: 10.1017/S0143385797084988.

[45]

B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geom., 20 (1998), 265-279.  doi: 10.1007/PL00009386.

[46] M. Viana, Lectures on Lyapunov Exponents, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781139976602.
[47]

A. Vince, Digit tiling of Euclidean space, In Directions in Mathematical Quasicrystals, M. Baake and R. V. Moody (eds.), AMS, Providence, RI (2000), 329–370. doi: 10.1112/s0024610700008711.

[48]

A. Vince, Rep-tiling Euclidean space, Aequationes Math., 50 (1995), 191-213.  doi: 10.1007/BF01831118.

show all references

References:
[1]

I. Abou and P. Liardet, Flots chaînés, Proceedings of the Sixth Congress of Romanian Mathematicians Vol.1, L. Beznea, V. Brinzanescu, R. Purice, et.al. (eds.), Editura Academiei Române, Bucharest (2009), 401–432.

[2]

J.-P. Allouche and P. Liardet, Generalized Rudin–Shapiro sequences, Acta Arith., 60 (1991), 1-27.  doi: 10.4064/aa-60-1-1-27.

[3]

M. BaakeN. P. FrankU. Grimm and E. A. Robinson, Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction, Studia Math., 247 (2019), 109-154.  doi: 10.4064/sm170613-10-3.

[4]

M. Baake, F. Gähler and U. Grimm, Examples of substitution systems and their factors, J. Int. Seq., 16 (2013), 13.2.14, 18 pp.

[5]

M. BaakeF. Gähler and N. Mañibo, Renormalisation of pair correlation measures for primitive inflation rules and absence of absolutely continuous diffraction, Commun. Math. Phys., 370 (2019), 591-635.  doi: 10.1007/s00220-019-03500-w.

[6] M. Baake and U. Grimm, Aperiodic Order Volume 1. A Mathematical Invitation, Cambridge University Press, Cambridge, 2013.  doi: 10.1017/CBO9781139025256.
[7]

M. Baake and U. Grimm, Squirals and beyond: Substitution tilings with singular continuous spectrum, Ergodic Th. & Dynam. Syst., 34 (2014), 1077-1102.  doi: 10.1017/etds.2012.191.

[8]

M. Baake and D. Lenz, Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra, Ergod. Th. & Dynam. Syst., 24 (2004), 1867-1893.  doi: 10.1017/S0143385704000318.

[9]

M. BaakeD. Lenz and A. van Enter, Dynamical versus diffraction spectrum for structures with finite local complexity, Ergod. Th. & Dynam. Syst., 35 (2015), 2017-2043.  doi: 10.1017/etds.2014.28.

[10]

E. Bannai and E. Bannai, Spin models on finite cyclic groups, J. Alg. Combin., 3 (1994), 243-259.  doi: 10.1023/A:1022407800541.

[11] L. Barreira and Y. Pesin, Nonuniform Hyperbolicity, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9781107326026.
[12]

A. Bartlett, Spectral theory of $\mathbb{Z}^d$ substitutions, Ergodic Th. & Dynam. Syst., 38 (2018), 1289-1341.  doi: 10.1017/etds.2016.66.

[13]

A. Berlinkov and B. Solomyak, Singular substitutions of constant length, Ergodic Th. & Dynam. Syst., 39 (2019), 2384-2402.  doi: 10.1017/etds.2017.133.

[14]

A. I. Bufetov and B. Solomyak, A spectral cocycle for substitution systems and translation flows, J. Anal. Math., 141 (2020), 165-205.  doi: 10.1007/s11854-020-0127-2.

[15]

C. Cabezas, Homomorphisms between multidimensional constant-shape substitutions, preprint, arXiv: 2106.10504.

[16]

L. ChanU. Grimm and I. Short, Substitution-based structures with absolutely continuous spectrum, Indag. Math., 29 (2018), 1072-1086.  doi: 10.1016/j.indag.2018.05.009.

[17]

M. I. Cortez, $ {\mathbb{Z}}^d$ Toeplitz arrays, Discr. Contin. Dynam. Syst. A, 15 (2006), 859-881.  doi: 10.3934/dcds.2006.15.859.

[18]

E. M. Coven and A. Meyerowitz, Tiling the integers with translates of one finite set, J. Algebra, 212 (1999), 161-174.  doi: 10.1006/jabr.1998.7628.

[19]

J. Dubédat, Topics on abelian spin models and related problems, Probab. Surveys, 8 (2011), 374-402.  doi: 10.1214/11-PS187.

[20]

E. H. el Abdalaoui and M. Lemańczyk, Approximately transitive dynamical systems and simple spectrum, Arch. Math., 97 (2011), 187-197.  doi: 10.1007/s00013-011-0285-7.

[21]

T. Fernique, Local rule substitutions and stepped surfaces, Theoret. Comp. Sci., 380 (2007), 317-329.  doi: 10.1016/j.tcs.2007.03.021.

[22]

N. P. Frank, Introduction to hierarchical tiling dynamical systems, In Substitution and Tiling Dynamics: Introduction to Self-inducing Structures, S. Akiyama and P. Arnoux (eds.), LNM 2773, Springer, Cham (2020), 33–95. doi: 10.1007/978-3-030-57666-0_2.

[23]

N. P. Frank, Multidimensional constant-length substitution sequences, Topology & Appl., 152 (2005), 44-69.  doi: 10.1016/j.topol.2004.08.014.

[24]

N. P. Frank, Substitution sequences in $\mathbb{Z}^d$ with a nonsimple Lebesgue component in the spectrum, Ergodic Th. & Dynam. Syst., 23 (2003), 519-532.  doi: 10.1017/S0143385702001256.

[25]

N. P. Frank and L. Sadun, Fusion: A general framework for hierarchical tilings of $\mathbb{R}^d$, Geom. Dedicata, 171 (2014), 149-186.  doi: 10.1007/s10711-013-9893-7.

[26]

N. P. Frank and L. Sadun, Fusion tilings with infinite local complexity, Top. Proc. 43 (2014) 235–276.

[27]

R. Greenfeld and T. Tao, The structure of translational tilings in $\mathbb{Z}^d$, Discr. Anal., 16 (2021), arXiv: 2010.03254, 28 pp. doi: 10.19086/da.

[28]

K. Gröchenig and A. Haas, Self-similar lattice tilings, J. Fourier Anal. Appl., 1 (1994), 131-170.  doi: 10.1007/s00041-001-4007-6.

[29]

P. R. Halmos and J. von Neumann, Operator methods in classical mechanics II, Ann. Math., 43 (1942), 332-350.  doi: 10.2307/1968872.

[30]

H. Helson, Cocycles on the circle, J. Oper. Theory, 16 (1986), 189-199. 

[31]

R. Kenyon, Self-replicating tilings, Contemp. Math., 135 (1992), 239-263.  doi: 10.1090/conm/135/1185093.

[32]

J. C. Lagarias and Y. Wang, Integral self-affine tiles in $\mathbb{R}^n$ I. standard and nonstandard digit sets, J. London Math. Soc., 54 (1996), 161-179.  doi: 10.1112/jlms/54.1.161.

[33]

J. C. Lagarias and Y. Wang, Self-affine tiles in $\mathbb{R}^n$, Adv. Math., 121 (1996), 21-49.  doi: 10.1006/aima.1996.0045.

[34]

J.-Y. LeeR. V. Moody and B. Solomyak, Consequences of pure point diffraction spectra for multiset substitution systems, Discrete Comput. Geom., 29 (2003), 525-560.  doi: 10.1007/s00454-003-0781-z.

[35]

J.-Y. LeeR. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra, Ann. Henri Poincaré, 3 (2002), 1003-1018.  doi: 10.1007/s00023-002-8646-1.

[36]

D. Lenz, Spectral theory of dynamical systems as diffraction theory of sampling functions, Monats. Math., 192 (2020), 625-649.  doi: 10.1007/s00605-020-01419-2.

[37]

N. Mañibo, Lyapunov exponents for binary substitutions of constant length, J. Math. Phys., 58 (2017), 113504, 9 pp. doi: 10.1063/1.4993169.

[38]

N. Mañibo, D. Rust and J. Walton, Spectral properties of substitutions on compact alphabets, preprint, arXiv: 2108.01762.

[39]

R. Meshulam, On subsets of finite abelian groups with no $3$-term arithmetic progression, J. Combin. Theor. A, 71 (1995), 168-172.  doi: 10.1016/0097-3165(95)90024-1.

[40]

M. G. Nadkarni, The skew product, In Spectral Theory of Dynamical Systems, R. B. Bapat, V. S. Borkar, P. Chaudhuri, et.al. (eds.), Hindustan Book Agency, Gurgaon (1998), 37–39.

[41]

M. Queffélec, Substitution Dynamical Systems–Spectral Analysis, 2nd. ed., LNM 1294, Springer, Berlin, 2010. doi: 10.1007/978-3-642-11212-6.

[42]

M. Queffélec, Une nouvelle properiété des suites de Rudin–Shapiro, Ann. Inst. Fourier, 37 (1987), 115-138.  doi: 10.5802/aif.1089.

[43]

E. A. Robinson, Non-abelian extensions have nonsimple spectrum, Compos. Math., 65 (1988), 155-170.

[44]

B. Solomyak, Dynamics of self-similar tilings, Ergodic Th. & Dynam. Syst., 17 (1997), 695–738 and Ergodic Th. & Dynam. Syst., 19 (1999), 1685 (erratum). doi: 10.1017/S0143385797084988.

[45]

B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geom., 20 (1998), 265-279.  doi: 10.1007/PL00009386.

[46] M. Viana, Lectures on Lyapunov Exponents, Cambridge University Press, Cambridge, 2014.  doi: 10.1017/CBO9781139976602.
[47]

A. Vince, Digit tiling of Euclidean space, In Directions in Mathematical Quasicrystals, M. Baake and R. V. Moody (eds.), AMS, Providence, RI (2000), 329–370. doi: 10.1112/s0024610700008711.

[48]

A. Vince, Rep-tiling Euclidean space, Aequationes Math., 50 (1995), 191-213.  doi: 10.1007/BF01831118.

Figure 1.  The supertiles $ \mathcal{S}^{1} ({\mathsf{a}}), \mathcal{S}^{2} ({\mathsf{a}}) $, and $ \mathcal{S}^{3} ({\mathsf{a}}) $ for Example 2
Figure 2.  The word $ (0,0),(0,-1), (-1,-1), (-1,0) \mapsto \, {\mathsf{a}},{\mathsf{a}},{\mathsf{b}},{\mathsf{a}} $ substituted six times. The presence of arbitrarily large rectangular words allows us to define a subshift of $ {\mathbb{Z}}^2 $ for $ {\mathcal{S}} $
Figure 3.  Three iterations of the pink $ {\mathsf{a}} $-tile located at the origin
Figure 4.  The image of some $ {\tau} \in {\Sigma} $ embedded into the tiling spaces $ {\Omega}_{{\mathfrak{u}}} $ (top) and $ {\Omega}_{{\mathfrak{t}}} $ (bottom)
Figure 5.  The alphabet. Spins are depicted as colors that vary in shade by digit
Figure 6.  The 1- and 2-supertiles for the triomino substitution, arranged by digit and spin as in Figure 5
Figure 7.  An indication that spins are uniformly distributed in the triomino subshift
Figure 8.  The level -1 and -2 supertiles of the Vierdrachen substitution
Figure 9.  The level-9 supertile for $ {\mathsf{d}}_0 $ and its image under the forget-the-spins and the forget-the-digits maps
Figure 10.  The image of $ {\mathcal{S}}^{9}({\mathsf{d}}_0) $ under the single-block codes given by the nontrivial characters. From (L) to (R), the corresponding character $ \chi $ and the spectral type of $ H^{\chi} $: $ \chi^{ }_1 $ ($\textsf{ac}$), $ \chi^{ }_2 $ ($\textsf{sc}$) and $ \chi^{ }_3 $ ($\textsf{ac}$)
Table 1.  Numerical values for upper bounds for $ 2f(N) $ for $ B_{\chi^{ }_1} $. Here $ \|\cdot\|^{ }_{\text{F}} $ stands for the Frobenius norm. All numerical errors are less than $ 10^{-3} $
$ N $ 10 11 12 13
$ \frac{1}{N}\int_{\mathbb{T}}\log\|B_{\chi^{ }_1}^{(N)}(\vec{x})\|^{2}_{\text{F}} $ 0.703953 0.695342 0.688005 0.682035
$ N $ 10 11 12 13
$ \frac{1}{N}\int_{\mathbb{T}}\log\|B_{\chi^{ }_1}^{(N)}(\vec{x})\|^{2}_{\text{F}} $ 0.703953 0.695342 0.688005 0.682035
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