# On the expressive power of quasiperiodic SFT

## TL;DR

This paper explores the structure and complexity of minimal and quasiperiodic shifts in symbolic dynamics, establishing new representations and complexity bounds for these classes of shift spaces.

## Contribution

It proves that effective minimal shifts can be represented as factors of projective subdynamics of minimal shifts of finite type in higher dimensions, extending Hochman's theorem.

## Key findings

- Every effective minimal shift is a factor of a projective subdynamics of a minimal SFT in higher dimension.
- A similar representation result holds for quasiperiodic shifts.
- Existence of a quasiperiodic SFT with patterns of Kolmogorov complexity (n) for size n.

## Abstract

In this paper we study the shifts, which are the shift-invariant and topologically closed sets of configurations over a finite alphabet in $\mathbb{Z}^d$. The minimal shifts are those shifts in which all configurations contain exactly the same patterns. Two classes of shifts play a prominent role in symbolic dynamics, in language theory and in the theory of computability: the shifts of finite type (obtained by forbidding a finite number of finite patterns) and the effective shifts (obtained by forbidding a computably enumerable set of finite patterns). We prove that every effective minimal shift can be represented as a factor of a projective subdynamics on a minimal shift of finite type in a bigger (by $1$) dimension. This result transfers to the class of minimal shifts a theorem by M.Hochman known for the class of all effective shifts and thus answers an open question by E.Jeandel. We prove a similar result for quasiperiodic shifts and also show that there exists a quasiperiodic shift of finite type for which Kolmogorov complexity of all patterns of size $n\times n$ is $\Omega(n)$.