Complexity as a homeomorphism invariant for tiling spaces. (La fonction de complexité comme invariant topologique des espaces de pavages.) (English. French summary) Zbl 1383.37013
The author studies aperiodic tilings with the following properties. In a tiling there are finitely many types of tiles up to translation. Any two tiles can be fitted together in finitely many ways, this condition is known as finite local complexity. A tiling is repetitive in the sense that any finite patch of it repeats within a bounded distance from any point of the tiling. An aperiodic tiling with the above properties has a topological compact space associated with it.
It is proved that if two aperiodic repetitive tilings with finite local complexity have homeomorphic tiling spaces, then their associated complexity functions are equivalent in a certain sense. This theorem can be reworded in terms of \(d\)-dimensional infinite words: if two \(\mathbb Z^d\)-subshifts (with the same conditions as above) are flow equivalent, their complexity functions are equivalent up to rescaling. An analogous theorem is stated for the repetitivity function, which is a quantitative measure of the recurrence of orbits in the tiling space.
Also it is shown that a certain cohomology group is an invariant of homeomorphisms between tiling spaces up to topological conjugacy.
It is proved that if two aperiodic repetitive tilings with finite local complexity have homeomorphic tiling spaces, then their associated complexity functions are equivalent in a certain sense. This theorem can be reworded in terms of \(d\)-dimensional infinite words: if two \(\mathbb Z^d\)-subshifts (with the same conditions as above) are flow equivalent, their complexity functions are equivalent up to rescaling. An analogous theorem is stated for the repetitivity function, which is a quantitative measure of the recurrence of orbits in the tiling space.
Also it is shown that a certain cohomology group is an invariant of homeomorphisms between tiling spaces up to topological conjugacy.
Reviewer: Elizaveta Zamorzaeva (Chişinău)
MSC:
| 37B50 | Multi-dimensional shifts of finite type, tiling dynamics (MSC2010) |
| 37C15 | Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems |
| 52C23 | Quasicrystals and aperiodic tilings in discrete geometry |
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