Algebraic and structural aspects on multidimensional symbolic dynamics and delone sets: Forced periodicity and local complexity

Abstract

Classically, multidimensional symbolic dynamics studies colorings of the integer grid of different dimensions with finitely many colors. In this thesis, a configuration is a coloring of either the d-dimensional integer grid or the d-dimensional Euclidean space. A configuration is periodic if it is equal to some of its translation by a non-zero vector. Delone sets are certain subsets of Euclidean spaces, and they form mathematical models for crystals and quasicrystals. In particular, Delone sets are identified with certain configurations on the Euclidean space with only two colors.

This thesis focuses on algebraic and structural aspects on multidimensional symbolic dynamics and Delone sets. In particular, the connection between forced periodicity and local complexity is studied. In our considerations, it is usually assumed that the alphabets of the configurations have some algebraic structure. A polynomial is an annihilator of a configuration if their discrete convolution is the zero configuration. We consider configurations that have non-trivial annihilators. In particular, periodic configurations have non-trivial annihilators – they are annihilated by a difference polynomial. It is known that if a configuration on the integer grid with integer coefficients has a non-trivial annihilator, then it is annihilated by a product of finitely many difference polynomials. Consequently, the periodic decomposition theorem states that such a configuration is a sum of finitely many periodic functions. However, these functions may not be configurations, that is, they may get infinitely many distinct values.

In this thesis, a certain family of configurations with non-trivial annihilators motivated by coding and graph theory is studied. We give new proofs for some known results on their forced periodicity. Also, some new results are proved. In addition, improvements of the periodic decomposition theorem are proved under some more involved assumptions. Finally, configurations on the Euclidean space and in particular Delone sets are considered. Known concepts and results are generalized to this setting, and some differences between these two settings are emphasized.