Complexity as a homeomorphism invariant for tiling spaces

Julien, Antoine

Annales de l'Institut Fourier, Tome 67 (2017) no. 2, pp. 539-577.

Il est prouvé dans cet article que deux pavages apériodiques et répétitifs dont les espaces de pavages sont homéomorphes ont des fonctions de complexité asymptotiquement équivalentes en un certain sens. Cela implique que lorsque les fonctions de complexité croissent polynomialement, l’exposant du terme dominant est préservé par homéomorphisme. Ce théorème peut s’énoncer en termes de mots infinis $d$-dimensionels : si deux sous-décalages indexés par ${\mathbb{Z}}^d$ (avec les mêmes conditions) sont « flot-équivalents » (c’est-à-dire que leurs suspensions sont homéomorphes), alors leurs fonctions de complexité sont équivalentes à changement d’échelle près. Un théorème analogue peut être énoncé pour la fonction de répétitivité, qui donne une mesure quantitative de la vitesse de récurrence des orbites dans l’espace de pavages. De manière sous-jacente, se trouve le fait que tout homéomorphisme entre espaces de pavages est induit par une déformation des tuiles. Dans la dernière section, on utilise cette observation pour montrer qu’un certain groupe de cohomologie fournit un invariant des homéomorphismes entre espaces de pavages à conjugation près.

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.

Classification : 37B50, 37B10
Keywords: aperiodic tilings, complexity, repetitivity, flow-equivalence, orbit-equivalence
Mot clés : pavages apériodiques, complexité, répétitivité, équivalence de flot, équivalence d’orbite

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