A Three-Dimensional SFT with Sparse Columns

We construct a nontrivial three-dimensional subshift of finite type whose projective $\Z$-subdynamics, or $\Z$-trace, is 2-sparse, meaning that there are at most two nonzero symbols in any vertical column. The subshift is deterministic in the direction of the subdynamics, so it is topologically conjugate to the set of spacetime diagrams of a partial cellular automaton. We also present a variant of the subshift that is defined by Wang cubes, and one whose alphabet is binary.

💡 Research Summary

This paper presents a groundbreaking construction in symbolic dynamics by building a three-dimensional subshift of finite type (SFT) that possesses a “sparse trace,” a property previously proven impossible in one and two dimensions. The authors’ main achievement is the explicit construction of a nontrivial 3D SFT where, in every valid configuration, each vertical column (aligned with the Z-axis) contains at most two nonzero symbols. This property is referred to as the SFT having a 2-sparse Z-projective subdynamics, or Z-trace.

The construction is geometrically motivated. The authors define a special class of three-dimensional surfaces called “mats.” A mat can be visualized as a fabric woven from parallel “strings” suspended between two rigid rods, where each string is a continuous curve in the (y,z)-plane for a fixed x-coordinate. A key designed property of these mats is “fluctuation”: while points on a mat share similar x and y coordinates, their z-coordinates can vary arbitrarily, meaning two horizontally close points can be vertically extremely far apart. The authors prove a Fundamental Lemma of Mats, which states that if two disjoint mats have identical horizontal projections, then one must lie entirely above or below the other. This geometric lemma is the cornerstone of the sparsity proof.

The 3D SFT, denoted X, is constructed via a substitution rule such that every non-zero point in any valid configuration can be interpreted as belonging to a single mat surface embedded in the lattice. The local rules of the SFT encode the geometry of the mat. The consequence of the Fundamental Lemma, when translated into the constraints of the SFT, is that it becomes impossible for three distinct points from the mat (and thus three nonzero symbols) to share the same (x,y) coordinate—the vertical column. A detailed analysis shows that each column can indeed contain at most two such points.

The paper establishes several key theorems. Theorem 1 quantifies the sparsity for a variant defined by Wang cubes, giving precise values for conjugacy-invariant sparsity measures (e.g., α(X)=2, β(X)=2). Theorem 2 shows that for a binary alphabet variant, the Z-trace is exactly the sofic shift X≤2, consisting of all bi-infinite sequences containing at most two 1s. Corollary 3 is particularly significant: since X≤2 is a countable sofic shift with a universal period, a known result by Pavlov and Schraudner implies it cannot be the trace of any two-dimensional SFT. This conclusively demonstrates that the class of possible trace subshifts is strictly richer for 3D SFTs than for 2D SFTs.

Furthermore, the constructed SFT is vertically deterministic, meaning the contents of any horizontal plane uniquely determine the plane below it. This makes it topologically conjugate to the spacetime diagrams of a partial two-dimensional cellular automaton (CA). Corollary 4 leverages this to show that this SFT admits a CA (specifically, a shift map) which is asymptotically nilpotent (all configurations converge to the all-zero state in the product topology) but not nilpotent (the convergence is not uniform in time). This resolves a question about the existence of such dynamics on higher-dimensional SFTs.

In summary, this work provides the first example of a nontrivial 3D SFT with a sparse trace, solving an open problem. It deepens the understanding of the relationships between projective subdynamics, determinism, cellular automata, and nilpotency in multidimensional symbolic dynamics.