Module Shifts and Measure Rigidity in Linear Cellular Automata

Suppose R is a finite commutative ring of prime characteristic, A is a finite Rmodule, M := Z×N , and Φ is an R-linear cellular automaton on A. If μ is a Φ-invariant measure which is multiply σ-mixing in a certain way, then we show that μ must be the Haar measure on a coset of some submodule shift of A. Under certain conditions, this means μ must be the uniform Bernoulli measure on A. Let A be a finite set. Let M := Z × N be a (D+E)-dimensional lattice, for some D,E ∈ N, and let A denote the set of all functions a : M−→A, which we regard as Mindexed configurations of elements in A. We write such a configuration as a = [am]m∈M, where am ∈ A for all m ∈ M. Treat A as a discrete topological space; then A is a Cantor space —i.e. it is compact, perfect, totally disconnected, and metrizable. If a ∈ A and U ⊂ M, then we define aU ∈ A by aU := [au]u∈U. If m ∈ M, then strictly speaking, am+U ∈ A; however, it will often be convenient to ‘abuse notation’ and treat am+U as an element of A U in the obvious way. Let H ⊂ M be some finite subset, and let φ : A−→A be a function (called a local rule). The cellular automaton (CA) determined by φ is the function Φ : A−→A defined by Φ(a)m = φ(am+H) for all a ∈ A and m ∈ M. We refer to H as the neighbourhood of Φ. We will prove a new ‘measure rigidity’ result for linear CA: if Φ is a linear CA and μ is a Φ-invariant measure which is multiply σ-mixing in a certain way, then μ must be the Haar measure on a coset of some submodule shift of A. In particular, if A admits no proper mixing subgroup shifts (e.g D = 1 and A = Z/p, for p prime), then μ must be the uniform measure on A. This result is complementary to previous rigidity results of [Sch95b, HMM03, Piv05, Ein05, Sab07]. Terminology & Notation. Throughout, lowercase bold-faced letters (a,b, c, . . .) denote elements ofA, and Roman letters (a, b, c, . . .) are elements ofA or ordinary numbers. Lowercase sans-serif (. . . ,m, n, p) are elements of M, and upper-case hollow font (U,V,W, . . .) are subsets of M. For any v ∈ M, let σ : A−→A be the shift map defined by σv(a)m = am+v for all a ∈ A and m ∈ M. Let CA(A) denote the set of cellular automata on A; then CA(A) is also the set of continuous transformations of A which commute with all shifts [Hed69, Theorem 3.4]. A subshift is a closed subset S ⊆ A which is invariant under all shifts. Let CA(S) := { Φ ∈ CA(A) ; Φ(S) ⊆ S } . If a ∈ A, and K ⊂ M, recall that aK := [ak]k∈K ∈ A. If S ⊆ A is a subshift, let SK := {sK ; s ∈ S} ⊆ A . If k ∈ A, then let 〈k〉 := { a ∈ A ; aK = k } be the cylinder set