arXiv subjects

Antoine Julien

Publications and source records attributed to Antoine Julien.

16 recordsLinked to original sources

arXiv2026

On balance properties of hypercubic billiard words

This paper studies balance properties for billiard words. Billiard words generalize Sturmian words by coding trajectories in hypercubic billiards. In the setting of aperiodic order, they also provide the simplest examples of quasicrystals, as tilings of the line obtained via cut and project sets with a cubical canonical window. By construction, the number of occurrences of each letter in a factor (i.e., a string of consecutive letters) of a hypercubic billiard word only depends on the length of the factor, up to an additive constant. In other words, the difference of the number of occurrences of each letter in factors of the same length is bounded. In contrast with the behaviour of letters, we prove the existence of words that are not balanced in billiard words: the difference of the number of occurrences of such unbalanced factors in longer factors of the same length is unbounded. The proof relies both on topological methods inspired by tiling cohomology and on arithmetic results on bounded remainder sets for toral translations.

math.DS↗

arXiv2019

Spectral triples and wavelets for higher-rank graphs

In this paper, we present a new way to associate a finitely summable spectral triple to a higher-rank graph $Λ$, via the infinite path space $Λ^\infty$ of $Λ$. Moreover, we prove that this spectral triple has a close connection to the wavelet decomposition of $Λ^\infty$ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the concept of stationary $k$-Bratteli diagrams, in order to associate a family of ultrametric Cantor sets, and their associated Pearson-Bellissard spectral triples, to a finite, strongly connected higher-rank graph $Λ$. We then study the zeta function, abscissa of convergence, and Dixmier trace associated to the Pearson-Bellissard spectral triples of these Cantor sets, and show these spectral triples are $ζ$-regular in the sense of Pearson and Bellissard. We obtain an integral formula for the Dixmier trace given by integration against a measure $μ$, and show that $μ$ is a rescaled version of the measure $M$ on $Λ^\infty$ which was introduced by an Huef, Laca, Raeburn, and Sims. Finally, we investigate the eigenspaces of a family of Laplace-Beltrami operators associated to the Dirichlet forms of the spectral triples. We show that these eigenspaces refine the wavelet decomposition of $L^2(Λ^\infty, M)$ which was constructed by Farsi et al.

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arXiv2018

Tiling deformations, cohomology, and orbit equivalence of tiling spaces

We study homeomorphisms of tiling spaces with finite local complexity (FLC), of which suspensions of $d$-dimensional subshifts are an example, and orbit equivalence of tiling spaces with (possibly) infinite local complexity (ILC). In the FLC case, we construct a cohomological invariant of homeomorphisms, and show that all homeomorphisms are a combination of tiling deformations, translations, and local equivalences (MLD). In the ILC case, we construct a cohomological invariant in the so-called weak cohomology, and show that all orbit equivalences are combinations of tiling deformations, translations, and topological conjugacies. These generalize results of Parry and Sullivan to higher dimensions. When the tiling spaces are uniquely ergodic, we show that homeomorphisms (FLC) or orbit equivalences (ILC) are completely parametrized by the appropriate cohomological invariants. We also show that, under suitable cohomological conditions, continuous maps between tiling spaces are homotopic to compositions of tiling deformations and either local derivations (FLC) or factor maps (ILC).

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arXiv2018

Spectral triples for higher-rank graph $C^*$-algebras

In this note, we present a new way to associate a spectral triple to the noncommutative $C^*$-algebra $C^*(Λ)$ of a strongly connected finite higher-rank graph $Λ$. We generalize a spectral triple of Consani and Marcolli from Cuntz-Krieger algebras to higher-rank graph $C^*$-algebras $C^*(Λ)$, and we prove that these spectral triples are intimately connected to the wavelet decomposition of the infinite path space of $Λ$ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. In particular, we prove that the wavelet decomposition of Farsi et al. describes the eigenspaces of the Dirac operator of this spectral triple.

math.OA↗

arXiv2017

Statistics of patterns in typical cut and project sets

In this article pattern statistics of typical cubical cut and project sets are studied. We give estimates for the rate of convergence of appearances of patches to their asymptotic frequencies. We also give bounds for repetitivity and repulsivity functions. The proofs use ideas and tools developed in discrepancy theory.

math.DS↗

arXiv2017

Wavelets and spectral triples for higher-rank graphs

In this paper, we present two new ways to associate a spectral triple to a higher-rank graph $Λ$. Moreover, we prove that these spectral triples are intimately connected to the wavelet decomposition of the infinite path space of $Λ$ which was introduced by Farsi, Gillaspy, Kang, and Packer in 2015. We first introduce the concept of stationary $k$-Bratteli diagrams, to associate a family of ultrametric Cantor sets to a finite, strongly connected higher-rank graph $Λ$. Then we show that under mild hypotheses, the Pearson-Bellissard spectral triples of such Cantor sets have a regular $ζ$-function, whose abscissa of convergence agrees with the Hausdorff dimension of the Cantor set, and that the measure $μ$ induced by the associated Dixmier trace agrees with the measure $M$ on the infinite path space $Λ^\infty$ of $Λ$ which was introduced by an Huef, Laca, Raeburn, and Sims. Furthermore, we prove that $μ= M$ is a rescaled version of the Hausdorff measure of the ultrametric Cantor set. From work of Julien and Savinien, we know that for $ζ$-regular Pearson-Bellissard spectral triples, the eigenspaces of the associated Laplace-Beltrami operator constitute an orthogonal decomposition of $L^2(Λ^\infty, μ)$; we show that this orthogonal decomposition refines the wavelet decomposition of Farsi et al. In addition, we generalize a spectral triple of Consani and Marcolli from Cuntz-Krieger algebras to higher-rank graph $C^*$-algebras, and prove that the wavelet decomposition of Farsi et al.~describes the eigenspaces of its Dirac operator.

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arXiv2016

K-theory of the Chair Tiling via AF-algebras

We compute the $K$-theory groups of the groupoid C$^\ast$-algebra of the chair tiling, using a new method. We use exact sequences of Putnam to compute these groups from the $K$-theory groups of the $AF$-algebras of the substitution and the induced lower dimensional substitutions on edges and vertices.

math.OA↗

arXiv2016

Wavelets and spectral triples for fractal representations of Cuntz algebras

In this article we provide an identification between the wavelet decompositions of certain fractal representations of $C^*-$algebras of directed graphs of M. Marcolli and A. Paolucci, and the eigenspaces of Laplacians associated to spectral triples constructed from Cantor fractal sets that are the infinite path spaces of Bratteli diagrams associated to the representations, with a particular emphasis on wavelets for representations of $\mathcal{O}_D$. In particular, in this setting we use results of J. Pearson and J. Bellissard, and A. Julien and J. Savinien, to construct first the spectral triple and then the Laplace Beltrami operator on the associated Cantor set. We then prove that in certain cases, the orthogonal wavelet decomposition and the decomposition via orthogonal eigenspaces match up precisely. We give several explicit examples, including an example related to a Sierpinski fractal, and compute in detail all the eigenvalues and corresponding eigenspaces of the Laplace Beltrami operators for the equal weight case for representations of Cuntz algebras, and in the uneven weight case for certain representations of $\mathcal{O}_2$, and show how the eigenspaces and wavelet subspaces at different levels are related.

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arXiv2015

Spectral triples for subshifts

We propose a construction for spectral triple on algebras associated with subshifts. One-dimensional subshifts provide concrete examples Z-actions on Cantor sets. The C*-algebra of this dynamical system is generated by functions in C(X) and a unitary element u implementing the action. Building on ideas of Christensen and Ivan, we give a construction of a family of spectral triples on the commutative algebra C(X). There is a canonical choice of eigenvalues for the Dirac operator D which ensures that [D,u] is bounded, so that it extends to a spectral triple on the crossed product. We study the summability of this spectral triple, and provide examples for which the Connes' distance associated with it on the commutative algebra is unbounded, and some for which it is bounded. We conjecture that our results on the Connes distance extend to the spectral triple defined on the noncommutative algebra.

math.OA↗

arXiv2014

On the noncommutative geometry of tilings

This is a chapter in an incoming book on aperiodic order. We review results about the topology, the dynamics, and the combinatorics of aperiodically ordered tilings obtained with the tools of noncommutative geometry.

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arXiv2014

Complexity as a homeomorphism invariant for tiling spaces

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. An analogue theorem is proved for the repetitivity function, which is a quantitative measure of the recurrence of orbits in the tiling space. How this result relates to the theory of tilings deformations is outlined in the last part.

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arXiv2013

Bi-Lipshitz Embedding of Ultrametric Cantor Sets into Euclidean Spaces

An ultrametric Cantor set can be seen as the boundary of a rooted weighted tree called the Michon tree. The notion of Assouad dimension is re-interpreted as seen on the Michon tree. The Assouad dimension of an ultrametric Cantor set is finite if and only if the space is bi-Lipschitz embeddable in a finite dimensional Euclidean space. This result, due to Assouad and refined by Luukkainen--Movahedi-Lankarani is re-proved in the Michon tree formalism. It is applied to answer the embedding question for some spaces which can be seen naturally as boundary of trees: linearly repetitive subshifts, Sturmian subshifts, and the boundary of Galton--Watson trees with random weights. Some of these give examples of nonembeddable spaces with finite Hausdorff dimension.

math.GN↗

arXiv2012

Combinatorics and topology of the Robinson tiling

We study the space of all tilings which can be obtained using the Robinson tiles (this is a two-dimensional subshift of finite type). We prove that it has a unique minimal subshift, and describe it by means of a substitution. This description allows to compute its cohomology groups, and prove that it is a model set.

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arXiv2010

Tiling groupoids and Bratteli diagrams II: structure of the orbit equivalence relation

In this second paper, we study the case of substitution tilings of R^d. The substitution on tiles induces substitutions on the faces of the tiles of all dimensions j=0, ..., d-1. We reconstruct the tiling's equivalence relation in a purely combinatorial way using the AF-relations given by the lower dimensional substitutions. We define a Bratteli multi-diagram B which is made of the Bratteli diagrams B^j, j=0, ..., d, of all those substitutions. The set of infinite paths in B^d is identified with the canonical transversal Xi of the tiling. Any such path has a "border", which is a set of tails in B^j for some j less than or equal to d, and this corresponds to a natural notion of border for its associated tiling. We define an etale equivalence relation R_B on B by saying that two infinite paths are equivalent if they have borders which are tail equivalent in B^j for some j less than or equal to d. We show that R_B is homeomorphic to the tiling's equivalence relation R_Xi.

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arXiv2009

Transverse Laplacians for Substitution Tilings

Pearson and Bellissard recently built a spectral triple - the data of Riemanian noncommutative geometry - for ultrametric Cantor sets. They derived a family of Laplace-Beltrami like operators on those sets. Motivated by the applications to specific examples, we revisit their work for the transversals of tiling spaces, which are particular self-similar Cantor sets. We use Bratteli diagrams to encode the self-similarity, and Cuntz-Krieger algebras to implement it. We show that the abscissa of convergence of the zeta-function of the spectral triple gives indications on the exponent of complexity of the tiling. We determine completely the spectrum of the Laplace-Beltrami operators, give an explicit method of calculation for their eigenvalues, compute their Weyl asymptotics, and a Seeley equivalent for their heat kernels.

math.OA↗

arXiv2008

Complexity and cohomology for cut and projection tilings

We consider a subclass of tilings, the tilings obtained by cut and projection. Under somewhat standard assumptions, we show that the natural complexity function has polynomial growth. We compute its exponent αin terms of the ranks of certain groups which appear in the construction. We give bounds for α. These computations apply to some well known tilings, such as the octagonal tilings, or tilings associated with billiard sequences. A link is made between the exponent of the complexity, and the fact that the cohomology of the associated tiling space is finitely generated over \Q. We show that such a link cannot be established for more general tilings, and we present a counter-example in dimension one.

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