Approximantsperiodic

Approximantsperiodic is a mathematical concept used in symbolic dynamics, combinatorics on words, and topological dynamics to describe infinite sequences or dynamical systems that can be closely approximated by periodic orbits. In this context, a sequence (or system) is called approximants periodic if for every given tolerance level ε>0, there exists a periodic sequence whose language – the set of all finite subwords that appear in it – lies within ε of the language of the original sequence in a suitable metric, such as the Hausdorff distance on languages. This property is stronger than being almost periodic; it requires that the approximating periodic sequences can be chosen with periods that grow arbitrarily large while still maintaining the required closeness.

The study of approximants periodic sequences is motivated by applications in the spectral theory of quasicrystals, the characterization of Sturmian and Arnoux–Rauzy sequences, and the classification of minimal dynamical systems. For example, every Sturmian sequence – sequences with minimal complexity exactly n+1 – can be approximated by periodic sequences with rational slopes that converge to the irrational slope of the Sturmian sequence. Similarly, substitution systems generated by primitive substitutions often admit a hierarchy of periodic approximations obtained by iterating the substitution a finite number of times and then repeating the finite block periodically. These approximations play a role in proving regularity results such as unique ergodicity, spectral gap properties, and in the construction of tiling spaces with Anderson–Putnam complexes.

The property is closely related to the notion of linear repetitivity and to concatenations of return words. A formal criterion: a minimal shift is approximants periodic if the collection of its return words has bounded complexity and each return word can be extended to a block that repeats with bounded gaps. This ensures that the shift’s orbit closure can be expressed as the inverse limit of finite cyclic covers. Researchers investigating quasicrystalline order, ergodic optimization, and the structure of tiling spaces frequently employ the approximants periodic framework as a tool to bridge the gap between finite periodic models and infinite aperiodic configurations.