Renormalisation for inflation tilings. II: Connections to number theory. (English) Zbl 1487.52026
Wood, David R. (ed.) et al., 2019–20 MATRIX annals. Cham: Springer. MATRIX Book Ser. 4, 709-711 (2021).
The paper is a two-pages expository (without proofs) of some results previously obtained by the author with Baake, Coons, Gaehler, and Grimm about some connections of the Lyapunov exponent of the matrix cocycles induced by Fourier matrices of inflation rules for aperiodic tilings and Mahler measures of certain polynomials.
For the entire collection see [Zbl 1459.37002].
For the entire collection see [Zbl 1459.37002].
Reviewer: Anton Shutov (Vladimir)
MSC:
| 52C23 | Quasicrystals and aperiodic tilings in discrete geometry |
| 37B52 | Tiling dynamics |
| 11J82 | Measures of irrationality and of transcendence |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 11R09 | Polynomials (irreducibility, etc.) |
| 42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
| 11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |
References:
| [1] | Baake, M., Coons, M., Mañibo, N.: Binary constant-length substitutions and Mahler measures of Borwein polynomials. In: Sims, B. (ed.) Proceedings of the Jonathan M. Borwein Commemorative Conference. Springer, Berlin, in press. http://arxiv.org/abs/1711.02492 · Zbl 1461.11143 |
| [2] | Baake, M., Frank, N.P., Grimm, U., Robinson, E.A.: Geometric properties of a binary non- Pisot inflation and absence of absolutely continuous diffraction. Studia Math. 247, 109-154 (2019) · Zbl 1419.37017 |
| [3] | Baake, M., G¨ahler, F., Mañibo, N.: Renormalisation of pair correlation measures for primitive inflation rules and absence of absolutely continuous diffraction. Preprint http://arxiv.org/abs/1805.09650 · Zbl 1433.37014 |
| [4] | Baake, M., Grimm, U.: Renormalisation of pair correlations and their Fourier transforms for primitive block substitutions. In: Akiyama, S., Arnoux, P. (eds.) Tiling and Discrete Geometry. Springer, Berlin, in press · Zbl 1457.52016 |
| [5] | Baake, M., Grimm, U., Mañibo, N.: Spectral analysis of a family of binary inflation rules. Lett. Math. Phys. 108, 1783-1805 (2018) · Zbl 1406.37014 |
| [6] | Mañibo, N.: Lyapunov exponents for binary constant-length substitutions. J. Math. Phys. 58, 113504:1-9 (2017) · Zbl 1380.37035 |
| [7] | Mañibo, N.: Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems. PhD thesis, Bielefeld University (2019) |
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