Squirals and Beyond Substitution Tilings With Singular Continuous Spectrum

[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. Baake, N. P. Frank, U. 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. Baake, F. 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. Baake, D. 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. Chan, U. 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.
[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.
[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. Lee, R. 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. Lee, R. 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.
[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.
[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).
[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.
[48]	A. Vince, Rep-tiling Euclidean space, Aequationes Math., 50 (1995) , 191-213.  doi: 10.1007/BF01831118.

woodruffbleturejaway.blogspot.com

Source: https://www.aimsciences.org/article/doi/10.3934/dcds.2022105

Share this post