Metallic mean Wang tiles II: the dynamics of an aperiodic computer chip
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2024
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| _version_ | 1866909805220724736 |
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| author | Labbé, Sébastien |
| author_facet | Labbé, Sébastien |
| contents | We consider a new family $(\mathcal{T}_n)_{n\geq1}$ of aperiodic sets of Wang tiles and we describe the dynamical properties of the set $Ω_n$ of valid configurations $\mathbb{Z}^2\to\mathcal{T}_n$. The tiles can be defined as the different instances of a square-shaped computer chip whose inputs and outputs are 3-dimensional integer vectors. The family include the Ammann aperiodic set of 16 Wang tiles and gathers the hallmarks of other small aperiodic sets of Wang tiles. Notably, the tiles satisfy additive versions of equations verified by the Kari--Culik aperiodic sets of 14 and 13 Wang tiles. Also configurations in $Ω_n$ are the codings of a $\mathbb{Z}^2$-action on a 2-dimensional torus like the Jeandel--Rao aperiodic set of 11 Wang tiles. The family broadens the relation between quadratic integers and aperiodic tilings beyond the omnipresent golden ratio as the dynamics of $Ω_n$ involves the positive root $β$ of the polynomial $x^2-nx-1$, also known as the $n$-th metallic mean. We show the existence of an almost one-to-one factor map $Ω_n\to\mathbb{T}^2$ which commutes with the shift action on $Ω_n$ with horizontal and vertical translations by $β$ on $\mathbb{T}^2$. The factor map can be explicitly defined by the average of the top labels from the same row of tiles as in Kari and Culik examples. The proofs are based on the minimality of $Ω_n$ (proved in a previous article) and a polygonal partition of $\mathbb{T}^2$ which we show is a Markov partition for the toral $\mathbb{Z}^2$-action. The partition and the sets of Wang tiles are symmetric which makes them, like Penrose tilings, worthy of investigation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_03197 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Metallic mean Wang tiles II: the dynamics of an aperiodic computer chip Labbé, Sébastien Dynamical Systems Combinatorics Metric Geometry Number Theory 52C23 (Primary) 37B51, 37A05, 11B39 (Secondary) We consider a new family $(\mathcal{T}_n)_{n\geq1}$ of aperiodic sets of Wang tiles and we describe the dynamical properties of the set $Ω_n$ of valid configurations $\mathbb{Z}^2\to\mathcal{T}_n$. The tiles can be defined as the different instances of a square-shaped computer chip whose inputs and outputs are 3-dimensional integer vectors. The family include the Ammann aperiodic set of 16 Wang tiles and gathers the hallmarks of other small aperiodic sets of Wang tiles. Notably, the tiles satisfy additive versions of equations verified by the Kari--Culik aperiodic sets of 14 and 13 Wang tiles. Also configurations in $Ω_n$ are the codings of a $\mathbb{Z}^2$-action on a 2-dimensional torus like the Jeandel--Rao aperiodic set of 11 Wang tiles. The family broadens the relation between quadratic integers and aperiodic tilings beyond the omnipresent golden ratio as the dynamics of $Ω_n$ involves the positive root $β$ of the polynomial $x^2-nx-1$, also known as the $n$-th metallic mean. We show the existence of an almost one-to-one factor map $Ω_n\to\mathbb{T}^2$ which commutes with the shift action on $Ω_n$ with horizontal and vertical translations by $β$ on $\mathbb{T}^2$. The factor map can be explicitly defined by the average of the top labels from the same row of tiles as in Kari and Culik examples. The proofs are based on the minimality of $Ω_n$ (proved in a previous article) and a polygonal partition of $\mathbb{T}^2$ which we show is a Markov partition for the toral $\mathbb{Z}^2$-action. The partition and the sets of Wang tiles are symmetric which makes them, like Penrose tilings, worthy of investigation. |
| title | Metallic mean Wang tiles II: the dynamics of an aperiodic computer chip |
| topic | Dynamical Systems Combinatorics Metric Geometry Number Theory 52C23 (Primary) 37B51, 37A05, 11B39 (Secondary) |
| url | https://arxiv.org/abs/2403.03197 |