Tilings, Quasicrystals, Discrete Planes, Generalized Substitutions, and Multidimensional Continued Fractions
نویسندگان
چکیده
The aim of this paper is to give an overview of recent results about tilings, discrete approximations of lines and planes, and Markov partitions for toral automorphisms. The main tool is a generalization of the notion of substitution. The simplest examples which correspond to algebraic parameters, are related to the iteration of one substitution, but we show that it is possible to treat arbitrary irrational examples by using multidimensional continued fractions. We give some non-trivial applications to Diophantine approximation, numeration systems and tilings, and we expose the main unsolved questions.
منابع مشابه
Arithmetic Discrete Planes Are Quasicrystals
Arithmetic discrete planes can be considered as liftings in the space of quasicrystals and tilings of the plane generated by a cut and project construction. We first give an overview of methods and properties that can be deduced from this viewpoint. Substitution rules are known to be an efficient construction process for tilings. We then introduce a substitution rule acting on discrete planes, ...
متن کاملFunctional stepped surfaces, flips, and generalized substitutions
A substitution is a non-erasing morphism of the free monoid. The notion of multidimensional substitution of non-constant length acting on multidimensional words introduced in [AI01,ABS04,Fer05a,Fer05b,Fer05c] is proved to be well-defined on the set of two-dimensional words related to discrete approximations of irrational planes. Such a multidimensional substitution can be associated with any us...
متن کاملGeneration and Recognition of Digital Planes Using Multi-dimensional Continued Fractions
This paper extends, in a multi-dimensional framework, pattern recognition techniques for generation or recognition of digital lines. More precisely, we show how the connection between chain codes of digital lines and continued fractions can be generalized by a connection between tilings and multi-dimensional continued fractions. This leads to a new approach for generating and recognizing digita...
متن کاملGeneration of discrete planes
We consider the problem of generation of discrete planes using generalized substitutions. We give sufficient conditions to be sure to generate all of a discrete plane by a sequence of substitutions; these conditions, however, are not easy to check, even on simple examples. One can build approximations of discrete planes in several ways, namely as stepped surfaces (unions of faces), as sets of v...
متن کاملAbout thin arithmetic discrete planes
Arithmetic discrete planes are sets of integer points located within a fixed bounded distance (called thickness) of a Euclidean plane. We focus here on a class of “thin” arithmetic discrete planes, i.e., on a class of arithmetic discrete planes whose thickness is smaller than the usual one, namely the so-called standard one. These thin arithmetic discrete planes have “holes” but we consider a t...
متن کامل