Complexity as a homeomorphism invariant for tiling spaces
[La fonction de complexité comme invariant topologique des espaces de pavages]
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 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 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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3091
Classification : 37B50, 37B10
Keywords: aperiodic tilings, complexity, repetitivity, flow-equivalence, orbit-equivalence
Mot clés : pavages apériodiques, complexité, répétitivité, équivalence de flot, équivalence d’orbite
Julien, Antoine 1

1 Institutt for Matematiske fag NTNU 7491 Trondheim (Norway)
@article{AIF_2017__67_2_539_0,
     author = {Julien, Antoine},
     title = {Complexity as a homeomorphism invariant for tiling spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {539--577},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {67},
     number = {2},
     year = {2017},
     doi = {10.5802/aif.3091},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3091/}
}
TY  - JOUR
AU  - Julien, Antoine
TI  - Complexity as a homeomorphism invariant for tiling spaces
JO  - Annales de l'Institut Fourier
PY  - 2017
SP  - 539
EP  - 577
VL  - 67
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.3091/
DO  - 10.5802/aif.3091
LA  - en
ID  - AIF_2017__67_2_539_0
ER  - 
%0 Journal Article
%A Julien, Antoine
%T Complexity as a homeomorphism invariant for tiling spaces
%J Annales de l'Institut Fourier
%D 2017
%P 539-577
%V 67
%N 2
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.3091/
%R 10.5802/aif.3091
%G en
%F AIF_2017__67_2_539_0
Julien, Antoine. Complexity as a homeomorphism invariant for tiling spaces. Annales de l'Institut Fourier, Tome 67 (2017) no. 2, pp. 539-577. doi : 10.5802/aif.3091. http://archive.numdam.org/articles/10.5802/aif.3091/

[1] Baake, Michael; Lenz, Daniel; Richard, Christoph Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies, Lett. Math. Phys., Volume 82 (2007) no. 1, pp. 61-77 | DOI

[2] Baake, Michael; Schlottmann, Martin; Jarvis, Peter D. Quasiperiodic tilings with tenfold symmetry and equivalence with respect to local derivability, J. Phys. A, Volume 24 (1991) no. 19, pp. 4637-4654 http://stacks.iop.org/0305-4470/24/4637 | DOI

[3] Bellissard, Jean; Benedetti, Riccardo; Gambaudo, Jean-Marc Spaces of tilings, finite telescopic approximations and gap-labeling, Comm. Math. Phys., Volume 261 (2006) no. 1, pp. 1-41 | DOI

[4] Bellissard, Jean; Herrmann, D. J. L.; Zarrouati, M. Hulls of aperiodic solids and gap labeling theorems, Directions in mathematical quasicrystals (CRM Monogr. Ser.), Volume 13, Amer. Math. Soc., Providence, RI, 2000, pp. 207-258

[5] Benedetti, Riccardo; Gambaudo, Jean-Marc On the dynamics of 𝔾-solenoids. Applications to Delone sets, Ergodic Theory Dynam. Systems, Volume 23 (2003) no. 3, pp. 673-691 | DOI

[6] Boulmezaoud, Housem; Kellendonk, Johannes Comparing different versions of tiling cohomology, Topology Appl., Volume 157 (2010) no. 14, pp. 2225-2239 | DOI

[7] Clark, Alex; Sadun, Lorenzo When shape matters: deformations of tiling spaces, Ergodic Theory Dynam. Systems, Volume 26 (2006) no. 1, pp. 69-86 | DOI

[8] Frank, Natalie; Sadun, Lorenzo Fusion tilings with infinite local complexity, Topology Proc., Volume 43 (2014), pp. 235-276

[9] Julien, Antoine Complexité des pavages apériodiques : calculs et interprétations, Université Lyon 1 (2009) (Ph. D. Thesis https://tel.archives-ouvertes.fr/tel-00466323v1)

[10] Julien, Antoine Complexity and cohomology for cut-and-projection tilings, Ergodic Theory Dynam. Systems, Volume 30 (2010) no. 2, pp. 489-523 | DOI

[11] Julien, Antoine; Sadun, Lorenzo Tiling deformations, cohomology, and orbit equivalence of tiling spaces (2015) (http://arxiv.org/abs/1506.02694)

[12] Julien, Antoine; Savinien, Jean Transverse Laplacians for substitution tilings, Comm. Math. Phys., Volume 301 (2011) no. 2, pp. 285-318 | DOI

[13] Kellendonk, Johannes Noncommutative geometry of tilings and gap labelling, Rev. Math. Phys., Volume 7 (1995) no. 7, pp. 1133-1180 | DOI

[14] Kellendonk, Johannes Pattern-equivariant functions and cohomology, J. Phys. A, Volume 36 (2003) no. 21, pp. 5765-5772 | DOI

[15] Kellendonk, Johannes Pattern equivariant functions, deformations and equivalence of tiling spaces, Ergodic Theory Dynam. Systems, Volume 28 (2008) no. 4, pp. 1153-1176 | DOI

[16] Kellendonk, Johannes; Putnam, Ian F. The Ruelle-Sullivan map for actions of n , Math. Ann., Volume 334 (2006) no. 3, pp. 693-711 | DOI

[17] Kellendonk, Johannes; Sadun, Lorenzo Meyer sets, topological eigenvalues, and Cantor fiber bundles, J. Lond. Math. Soc., Volume 89 (2014) no. 1, pp. 114-130 | DOI

[18] Lagarias, Jeffrey C.; Pleasants, Peter A. B. Repetitive Delone sets and quasicrystals, Ergodic Theory Dynam. Systems, Volume 23 (2003) no. 3, pp. 831-867 | DOI

[19] Lenz, Daniel Aperiodic Linearly Repetitive Delone Sets Are Densely Repetitive, Discrete Comput Geom, Volume 31 (2004) no. 2, pp. 323-326 http://link.springer.com/article/10.1007/s00454-003-2903-z | DOI

[20] Morse, Marston; Hedlund, Gustav A. Symbolic Dynamics, Amer. J. Math., Volume 60 (1938) no. 4, pp. 815-866 | DOI

[21] Muhly, Paul S.; Renault, Jean N.; Williams, Dana P. Equivalence and isomorphism for groupoid C * -algebras, J. Operator Theory, Volume 17 (1987) no. 1, pp. 3-22

[22] Rand, Betseygail; Sadun, Lorenzo An approximation theorem for maps between tiling spaces, Discrete Contin. Dyn. Syst., Volume 29 (2011) no. 1, pp. 323-326 | DOI

[23] Renault, Jean A groupoid approach to C * -algebras, Lecture Notes in Mathematics, 793, Springer, Berlin, 1980, ii+160 pages

[24] Sadun, Lorenzo; Williams, R. F. Tiling spaces are Cantor set fiber bundles, Ergodic Theory Dynam. Systems, Volume 23 (2003) no. 1, pp. 307-316 | DOI

[25] Schlottmann, Martin Periodic and quasi-periodic Laguerre tilings, Internat. J. Modern Phys. B, Volume 7 (1993) no. 6-7, pp. 1351-1363 | DOI

Cité par Sources :