On the long-range order of the Spectre tilings
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912131142647808 |
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| author | Baake, Michael Gähler, Franz Mazáč, Jan Sadun, Lorenzo |
| author_facet | Baake, Michael Gähler, Franz Mazáč, Jan Sadun, Lorenzo |
| contents | The Spectre is an aperiodic monotile for the Euclidean plane that is truly chiral in the sense that it tiles the plane without any need for a reflected tile. The topological and dynamical properties of the Spectre tilings are very similar to those of the Hat tilings. Specifically, the Spectre sits within a complex $2$-dimensional family of tilings, most of which involve two shapes rather than one. All tilings in the family give topologically conjugate dynamics, up to an overall rescaling and rotation. They all have pure point dynamical spectrum with continuous eigenfunctions and may be obtained from a $4:2$ dimensional cut-and-project scheme with regular windows of Rauzy fractal type. The diffraction measure of any Spectre tiling is pure point as well. For fixed scale and orientation, varying the shapes is MLD equivalent to merely varying the projection direction. These properties all follow from the first Čech cohomology being as small as it possibly could be, leaving no room for shape changes that alter the dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15503 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the long-range order of the Spectre tilings Baake, Michael Gähler, Franz Mazáč, Jan Sadun, Lorenzo Dynamical Systems Metric Geometry 52C20, 37D40, 55N05, 52C23 The Spectre is an aperiodic monotile for the Euclidean plane that is truly chiral in the sense that it tiles the plane without any need for a reflected tile. The topological and dynamical properties of the Spectre tilings are very similar to those of the Hat tilings. Specifically, the Spectre sits within a complex $2$-dimensional family of tilings, most of which involve two shapes rather than one. All tilings in the family give topologically conjugate dynamics, up to an overall rescaling and rotation. They all have pure point dynamical spectrum with continuous eigenfunctions and may be obtained from a $4:2$ dimensional cut-and-project scheme with regular windows of Rauzy fractal type. The diffraction measure of any Spectre tiling is pure point as well. For fixed scale and orientation, varying the shapes is MLD equivalent to merely varying the projection direction. These properties all follow from the first Čech cohomology being as small as it possibly could be, leaving no room for shape changes that alter the dynamics. |
| title | On the long-range order of the Spectre tilings |
| topic | Dynamical Systems Metric Geometry 52C20, 37D40, 55N05, 52C23 |
| url | https://arxiv.org/abs/2411.15503 |