arXiv subjects

Ayşe Yıltekin-Karataş

Publications and source records attributed to Ayşe Yıltekin-Karataş.

2 recordsLinked to original sources

arXiv2026

Cut and project schemes in the Poincaré disc: From cocompact Fuchsian groups to chaotic Delone sets

A question raised by Davies et al [Phys. Rev. Lett. 131, 2023] is: "Can developing new cut and project models, where the lattice is not square or the curve is non-linear, generate better performing graded metamaterials?" In this article, we study a natural construction of such a cut and project scheme, namely, cut and project schemes in relation to cocompact Fuchsian groups acting on the Poincaré disc model of hyperbolic space. We present a condition on the fundamental domain (a hyperbolic polygon) of the group so that the resulting cut and project set $S \subset \mathbb{R}$ is a chaotic Delone set. We also investigate the set of tile lengths of $S$, namely $\mathcal{L}_{S} = \{ z - y : z,y \in S, \, z > y \; \text{and} \; (y,z) \cap S = \emptyset \}$, and show that this set is countably infinite. Finally, we apply our results to cocompact Fuchsian triangle groups and show that the resulting cut and project sets are chaotic Delone, complementing and extending the work of López et al. [Discrete Contin. Dyn. Syst. 41, 2021].

math.DS↗

arXiv2024

Continuous deformation of the Bowen-Series map associated to a cocompact triangle group

In 1979, for each signature for Fuchsian groups of the first kind, Bowen and Series constructed an explicit fundamental domain for one group of the signature, and from this a function on $\mathbb S^1$ tightly associated with this group. In general, their fundamental domain enjoys what has since been called the `extension property'. We determine the exact set of signatures for cocompact triangle groups for which this extension property can hold for any convex fundamental domain, and verify that for this restricted set, the Bowen-Series fundamental domain does have the property. To each Bowen-Series function in this corrected setting, we naturally associate four continuous deformation families of circle functions. We show that each of these functions is aperiodic if and only if it is surjective; and, is finite Markov if and only if its natural parameter is a hyperbolic fixed point of the triangle group at hand.

math.DS↗