Irreducibility and periodicity in \(\mathbb Z^2\) symbolic systems

Irreducibility and periodicity in \(\mathbb{Z}^{2}\) symbolic systems, Discrete Analysis 2025:17, 37 pp.

One of the central aims of symbolic dynamics is to understand how local rules governing configurations lead to global dynamical behaviour. In the one-dimensional case, this picture is fairly well understood. Strongly irreducible (SI) systems, which can be thought of as models with a robust mixing property, form a particularly important class. Roughly speaking, strong irreducibility means that any two finite patterns that appear in the system can always be combined into a single configuration, provided they are placed far enough apart. A classical theorem, proved independently by Bertrand and by Weiss, shows that such systems necessarily contain dense periodic points. Thus, in one dimension SI systems exhibit both mixing-type behaviour and periodic structure.

In two dimensions the situation is more delicate. Lightwood proved that SI shifts of finite type (those defined by finitely many forbidden patterns) always contain periodic points, suggesting that perhaps the one-dimensional theory continued to hold. But for general \(\mathbb{Z}^2\)-subshifts the question remained open: does strong irreducibility alone force periodicity?

This paper gives a decisive answer. Hochman constructs a strongly irreducible \(\mathbb{Z}^2\)-subshift that has no periodic points at all. The construction is intricate, since even small perturbations tend to introduce periodicity, but the broad idea is to build in many local “witnesses” of aperiodicity (small patterns that by themselves already prevent global periodicity) and distribute them in large geometric patterns in such a way that they can always be combined without conflict. The outcome is a system that retains the mixing property of strong irreducibility but avoids global periodic structure altogether.

The result closes a long-standing problem and highlights a striking contrast between one and two dimensions: properties that are tightly linked in one setting can pull apart in the other. It also brings new techniques to the subject, combining combinatorial and geometric ideas that may have further applications. The paper concludes with a number of open questions, the most significant of which asks whether strongly irreducible \(\mathbb{Z}^3\) shifts of finite type must contain periodic points. Another direction is to ask whether strongly irreducible sofic shifts (factors of finite-type shifts) in two dimensions can have no periodic points. Finding such examples would represent a major breakthrough.