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Antoine Julien
Complexity as a homeomorphism invariant for tiling spaces
(La fonction de complexité comme invariant topologique des espaces de pavages)
Annales de l'institut Fourier, 67 no. 2 (2017), p. 539-577, doi: 10.5802/aif.3091
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Class. Math.: 37B50, 37B10
Keywords: aperiodic tilings, complexity, repetitivity, flow-equivalence, orbit-equivalence

Résumé - Abstract

It is proved that whenever two aperiodic repetitive tilings with finite local complexity have homeomorphic tiling spaces, their associated complexity functions are asymptotically equivalent in a certain sense (which implies, if the complexity is polynomial, that the exponent of the leading term is preserved by homeomorphism). 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. Behind this result is the fact that any homeomorphism between tiling spaces is described by a so-called “shape deformation”. In the last section, we use this observation to show that a certain cohomology group is an invariant of homeomorphisms between tiling spaces up to topological conjugacy.

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