This paper investigates dynamical systems arising from the
 action by
 translations on the orbit closures of self-similar and
 self-affine tilings
 of . The main focus is on spectral properties of such
 systems
 which are shown to be uniquely ergodic. We establish
 criteria for
 weak mixing and pure discrete spectrum for wide classes of
 such systems.
 They are applied to a number of examples which include
 tilings with
 polygonal and fractal tile boundaries; systems with pure
 discrete,
 continuous and mixed spectrum.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0143385797084988

Dynamics of self-similar tilings

We prove that the dynamical system generated by a primitive unimodular substitution of the Pisot type ond letters satisfying a combinatorial condition which is easy to check, is measurably isomorphic to a domain exchange in R d 1 , and is a nite extension of a translation on the torus T d 1 I n the course of the proof, we introduce some potentially useful notions: the linear maps associated to a substitution and their dual maps, and the -structure for a dynamical system with respect to a pair of partitions

/pdf/pisot-substitutions-and-rauzy-fractals-44wipdxq2n.pdf

Pisot substitutions and Rauzy fractals

CONTENTS Introduction § 1. A short review of results § 2. Criteria for Fuchsian systems with regular singular points § 3. Preliminary information. The method of solution § 4. The solubility of the Riemann-Hilbert problem for an irreducible representation of dimension three § 5. Fuchsian systems of two equations on the Riemann sphere. The Fuchsian weight of a representation § 6. The Riemann-Hilbert problem for a system of three equations § 7. The Fuchsian weight and two-dimensional holomorphic vector bundles on the Riemann sphere Conclusion References

https://iopscience.iop.org/article/10.1070/RM1990v045n02ABEH002350/pdf

The Riemann-Hilbert problem

Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences

This paper surveys different constructions and properties of some multiple tilings (that is, finite-to-one coverings) of the space that can be associated with beta-numeration and substitutions. It is indeed possible, generalizing Rauzy’s and Thurston’s constructions, to associate in a natural way either with a Pisot number β (of degree d) or with a Pisot substitution σ (on d letters) some compact basic tiles that are the closure of their interior, that have non-zero measure and a fractal boundary; they are attractors of some graph- directed Iterated Function System. We know that some translates of these prototiles under a Delone set Γ (provided by β or σ) cover Rd−1; it is conjectured that this multiple tiling is indeed a tiling (which might be either periodic or self-replicating according to the translation set Γ). This conjecture is known as the Pisot conjecture and can also be reformulated in spectral terms: the associated dynamical systems have pure discrete spectrum. We detail here the known constructions for these tilings, their main properties, some applications, and focus on some equivalent formulations of the Pisot conjecture, in the theory of quasicrystals for instance. We state in particular for Pisot substitutions a finiteness property analogous to the well-known (F) property in beta-numeration, which is a sufficient condition to get a tiling.

https://hal-lirmm.ccsd.cnrs.fr/lirmm-00105299/document

Tilings associated with beta-numeration and substitutions.

/pdf/modified-jacobi-perron-algorithm-and-generating-markov-415108fhio.pdf

Modified Jacobi-Perron Algorithm and Generating Markov Partitions for Special Hyperbolic Toral Automorphisms

/pdf/a-residue-formula-for-chern-classes-associated-with-2ad2psydzp.pdf

A Residue Formula for Chern Classes Associated with Logarithmic Connections

/pdf/parallelogram-tilings-and-jacobi-perron-algorithm-2jcmjb8y6u.pdf

Parallelogram Tilings and Jacobi-Perron Algorithm

/pdf/on-the-number-of-apparent-singularities-of-a-linear-48bt4olbn3.pdf

On the Number of Apparent Singularities of a Linear Differential Equation

/pdf/on-the-multiple-gamma-functions-101t6mi600.pdf

Makoto Ohtsuki

Papers

Modified Jacobi-Perron Algorithm and Generating Markov Partitions for Special Hyperbolic Toral Automorphisms

A Residue Formula for Chern Classes Associated with Logarithmic Connections

Parallelogram Tilings and Jacobi-Perron Algorithm

On the Number of Apparent Singularities of a Linear Differential Equation

On the Multiple Gamma-Functions