A remark on inverse limits of effective subshifts

We show that, for every finitely generated group with decidable word problem and undecidable domino problem, there exists a sequence of effective subshifts whose inverse limit is not the topological factor of any effective dynamical system. This follows from considerations on the universality under topological factors for this class of dynamical systems.

💡 Research Summary

The paper investigates the relationship between effective dynamical systems (EDS) and weakly effective dynamical systems (wEDS) for finitely generated groups, focusing on the role of inverse limits of effective subshifts. An EDS is defined as a group action that can be described algorithmically: the underlying compact space is recursively compact and each generator of the group acts by a computable map. Effective subshifts—closed, shift‑invariant subsets of a full shift defined by a recursively enumerable set of forbidden patterns—are canonical examples of EDS.

A key observation, already known from earlier work, is that the class of EDS is not closed under topological factor maps: a factor of an EDS need not be an EDS. However, any topological factor of an EDS can be represented as an inverse limit of (not necessarily uniform) effective subshifts. This motivates the definition of wEDS: a system that is topologically conjugate to an inverse limit of effective subshifts (equivalently, an inverse limit of EDS). By construction wEDS are closed under factors, and every EDS is a wEDS.

The authors introduce the notion of universality. An EDS (resp. wEDS) is universal if every other EDS (resp. wEDS) of the same group admits a continuous, equivariant surjection onto it. Two central universality results are proved.

Theorem B (Universality of EDS). For a finitely generated group G with decidable word problem, a universal EDS exists if and only if the domino problem for G is decidable. The domino problem asks whether, given a finite set of forbidden patterns, the associated subshift of finite type (SFT) is non‑empty. The forward direction is proved by constructing a “master” effective subshift that simultaneously encodes all recursively enumerable pattern sets; the resulting system can factor onto any effective subshift, provided the domino problem is decidable (so emptiness can be detected). The converse shows that if a universal EDS existed, one could decide emptiness of any SFT by checking whether the universal system factors onto it, thereby solving the domino problem. This extends Hochman’s results for ℤ^d to arbitrary finitely generated groups with decidable word problem.

Theorem C (Universality of wEDS). For every finitely generated group G (no decidability hypothesis needed) there exists a universal wEDS. The construction enumerates all effective subshifts, forms their inverse limit, and shows that any wEDS can be obtained as a factor of this limit. Uniformity of the enumeration is not required, which allows the result to hold even when the domino problem is undecidable.

Combining these two theorems yields the main result, Theorem A: if G has decidable word problem but undecidable domino problem (e.g., G = ℤ^2), then there exists a wEDS that is not a topological factor of any EDS. The argument is a simple contradiction: assuming every wEDS factors onto some EDS would make the universal wEDS a factor of a universal EDS, contradicting Theorem B because the domino problem is undecidable.

The paper supplies detailed preliminaries: definitions of recursive enumerability, decidable word problem, effective closed sets, computable maps on Cantor space, and the Curtis‑Hedlund‑Lyndon theorem for subshifts. It proves that an effective subshift is exactly a subshift whose forbidden patterns form a recursively enumerable set (Proposition 2.8) and that a zero‑dimensional EDS can be represented as a shift on an effectively closed, shift‑invariant subset of {0,1}^{G×ℕ} (Proposition 2.9).

In Section 3 the authors prove Proposition 3.1 (existence of a universal EDS when the domino problem is decidable) by constructing a recursive enumeration of all recursively enumerable pattern families and embedding each into a single large subshift. Proposition 3.2 (converse) shows that a universal EDS would give an algorithm for the domino problem, using the fact that emptiness of an SFT can be reduced to the existence of a factor map from the universal system.

Section 4 establishes the existence of a universal wEDS for any finitely generated group. The construction uses the inverse limit of the sequence of all effective subshifts, with factor maps given by the natural projections. The authors verify that this inverse limit is indeed a wEDS and that any other wEDS admits a factor map onto it.

Section 5 discusses the remaining case of groups with decidable domino problem, suggesting that in such groups every wEDS might actually be a factor of some EDS, and outlines possible directions for a full characterization of topological factors of EDS.

Overall, the paper clarifies the landscape of computable symbolic dynamics: while effective subshifts can be combined via inverse limits to produce very rich dynamical systems (wEDS), the additional requirement that a system itself be an EDS imposes a strict computability constraint tied to the domino problem. The results demonstrate that inverse limits of effective subshifts can lie strictly outside the class of EDS, answering a question raised in earlier work and highlighting a subtle boundary between computability and topological dynamics.