Asymptotic randomization of sofic shifts by linear cellular automata

Let M = Z be a D-dimensional lattice, and let (A,+) be an abelian group. AM is then a compact abelian group under componentwise addition. A continuous function : AM −→ AM is called a linear cellular automaton if there is a finite subset F ⊂ M and non-zero coefficients φf ∈ Z so that, for any a ∈ AM, (a) = ∑ f∈F φf · σ f(a). Suppose that μ is a probability measure onAM whose support is a subshift of finite type or sofic shift. We provide sufficient conditions (on and μ) under which asymptotically randomizes μ, meaning that wk∗ − limJ j→∞ μ = η, where η is the Haar measure on AM, and J ⊂ N has Cesàro density one. In the case when = 1+ σ and A = (Z/p) (p prime), we provide a condition on μ that is both necessary and sufficient. We then use this to construct zero-entropy measures which are randomized by 1+ σ . 0. Introduction Let D ≥ 1, and let M := Z be the D-dimensional lattice. If A is a (discretely topologized) finite set, then AM is compact in the Tychonoff topology. For any v ∈ M, let σ v : AM−→AM be the shift map: σ v(a) := [bm|m∈M], where bm := am−v, for all m ∈ M. A cellular automaton (CA) is a continuousmap : AM−→AM which commutes with all shifts: for any m ∈ M, σm ◦ = ◦σm. Let η be the uniform Bernoulli measure on AM. If μ is another probability measure on AM, we say asymptotically randomizes μ if wk∗ − limJ j→∞ μ = η, where J ⊂ N has Cesàro density one. If (A,+) is a finite abelian group, then AM is a product group, and η is the Haar measure. A linear cellular automaton (LCA) is a CA with a finite subset F ⊂ M (with # (F) ≥ 2), and non-zero coefficients φf ∈ Z (for all f ∈ F) so that, for any a ∈ AM, (a) = ∑ f∈F φf · σ f(a). (1) LCA are known to asymptotically randomize a wide variety of measures [MHM03, MM98, MM99, Lin84, FMMN00], including those satisfying a correlation-decay 1178 M. Pivato and R. Yassawi condition called harmonic mixing [PY02, PY04, MMPY06]. However, all known sufficient conditions for asymptotic randomization (and for harmonicmixing, in particular) require μ to have full support, i.e. supp(μ) = AM. Here we investigate asymptotic randomization when supp(μ) AM. In particular, we consider the case when supp(μ) is a sofic shift or subshift of finite type. In §1, we let A = Z/p (p prime), and demonstrate asymptotic randomization for any Markov random field that is locally free, a much weaker assumption than full support. However, in §2 we show that harmonic mixing is a rather restrictive condition, by exhibiting a measure whose support is a mixing sofic shift but which is not harmonically mixing. Thus, in §3, we introduce the less restrictive concept of dispersion mixing (formeasures) and the dual concept of dispersion (for LCA), and state our main result: any dispersive LCA asymptotically randomizes any dispersion mixing measure. In §4, we letA = (Z/p) (p prime, s ∈ N) and introduce bipartite LCA, a broad class exemplified by the automaton 1+ σ . We then show that any bipartite LCA is dispersive. In §5, we show that any uniformly mixing and harmonically bounded measure is dispersion mixing. In particular, in §6, we show that this implies that any mixing Markov measure (supported on a subshift of finite type), and any continuous factor of a mixing Markov measure (supported on a sofic shift) is dispersion mixing and, thus, is asymptotically randomized by any dispersive LCA (e.g. 1 + σ ). Thus, the example of §2 is asymptotically randomized, even though it is not harmonically mixing. In §7, we refine the results of §§3 and 4 by introducing Lucas mixing (a weaker condition than dispersion mixing). When A = (Z/p) , we show that a measure is asymptotically randomized by the automaton 1 + σ if and only if it is Lucas mixing. Finally, in §8, we use Lucas mixing to construct a class of zero-entropy measures which are asymptotically randomized by 1+ σ , thereby refuting the conjecture that positive entropy is necessary for asymptotic randomization. Preliminaries and notation. Throughout, (A,+) is an abelian group (usually A = (Z/p) , where p is prime and s ∈ N). Elements of AM are denoted by boldfaced letters (e.g. a, b, c), and subsets by gothic letters (e.g. A, B, C). Elements of M are sans serif (e.g. l, m, n) and subsets are U,V,W. If U ⊂ M and a ∈ AM, then aU := [au|u∈U] is the ‘restriction’ of a to an element of AU. For any b ∈ AU, let [b] := {c ∈ AM; cU = b} be the corresponding cylinder set. In particular, if a ∈ AM, then [aU] := {c ∈ AM; cU = aU}. Measures. LetM(AM) be the set of Borel probability measures onAM. Ifμ ∈ M(AM) and I ⊂ M, then let μI ∈ M(AI) be the marginal projection of μ onto AI. If J ⊂ M and b ∈ AJ, then let μ(b) ∈ M(AM) be the conditional probability measure in the cylinder set [b]. In other words, for any X ⊂ AM, μ(b)[X] := μ(X ∩ [b])/μ[b]. In particular, if I ⊂ M is finite, then μ I ∈ M(AM) is the conditional probability measure on the I coordinates: for any c ∈ AI, μ I [c] := μ([c] ∩ [b])/μ[b]. Randomization of sofic shifts by linear cellular automata 1179 Subshifts. A subshift [Kit98, LM95] is a closed, shift-invariant subset X ⊂ AM. If U ⊂ M, then let XU := {xU; x ∈ X} be all admissible U-blocks in X. If U ⊂ M is finite, and W = {w1, . . . ,wN } ⊂ AU is a collection of admissible blocks, then the induced subshift of finite type (SFT) is the largest subshift X ⊂ AM such that XU = W. In other words, X := ⋂m∈M σm[W], where [W] := {a ∈ AM; aU ∈ W}. A sofic shift is the image of an SFT under a block map. In particular, ifM = Z and U = {0, 1}, then X is called a topological Markov shift, and the transition matrix of X is the matrix P = [pab]a,b∈A, where pab = 1 if [ab] ∈ W, and pab = 0 if [ab] ∈ W. Characters. Let T1 ⊂ C be the circle group. A character of AM is a continuous homomorphism χ : AM−→T1; the group of such characters is denoted by ÂM. For any χ ∈ ÂM there is a finite subset K ⊂ M, and non-trivial χk ∈  for all k ∈ K, such that, for any a ∈ AM, χ(a) = ∏k∈K χk(ak). We indicate this by writing ‘χ = ⊗k∈K χk’. The rank of χ is the cardinality of K. Cesàro density. If , n ∈ Z, then let [ . . . n) := {m ∈ Z; ≤ m < n}. If J ⊂ N, then the Cesàro density of J is defined as density (J) := lim N→∞ 1 N # (J ∩ [0 . . .N)). If J,K ⊂ N, then their relative Cesàro density is defined as rel density[J/K] := lim N→∞ # (J ∩ [0 . . .N)) # (K ∩ [0 . . .N)) . In particular, density (J) = rel density[J/N]. 1. Harmonic mixing of Markov random fields Let B ⊂ M be a finite subset, symmetric under multiplication by −1 (usually, B = {−1, 0, 1}D). For any U ⊂ M, we define cl(U) := {u + b; u ∈ U and b ∈ B} and ∂U := cl(U) \ U. For example, if M = Z and B = {−1, 0, 1}, then ∂{0} = {±1}. Let μ ∈ M(AM). Suppose U ⊂ M, and let V := ∂U and W = M \ cl(U). If b ∈ AV, then we say that b isolates U from W if the conditional measure μ(b) is a product of μ U and μ W . That is, for any U ⊂ AU and W ⊂ AW, we have μ(b)(U ∩ W) = μ U (U) · μ W (W). We say that μ is a Markov random field [Bré99, KS80] with interaction range B (or write ‘μ is a B-MRF’) if, for any U ⊂ M with V = ∂U and W = M \ cl(U), any choice of b ∈ AV isolates U fromW. For example, if M = Z and B = {−1, 0, 1}, then μ is a B-MRF iff μ is a (one-step) Markov chain. If B = [−N . . .N], then μ is a B-MRF iff μ is an N-step Markov chain. LEMMA 1.1. If μ is a Markov random field, then supp(μ) is a subshift of finite type. 1180 M. Pivato and R. Yassawi For example, if μ is a Markov chain onAZ, then supp(μ) is a topologicalMarkov shift. Let B ⊂ M, and let μ ∈ M(AM) be B-MRF. Let S := B \ {0}. For any b ∈ AS, let μ 0 ∈ M(A) be the conditional probability measure on the zeroth coordinate. We say that μ is locally free if, for any b ∈ AS, # (supp(μ 0 )) ≥ 2. Example. If D = 1, then B = {−1, 0, 1}, S = {±1}, and μ is a Markov chain. Thus, supp(μ) is a topological Markov shift, with transition matrix P = [pab]a,b∈A. For any a, b ∈ A, write a b if pab = 1, and define the follower and predecessor sets F(a) := {b ∈ A; a b} and P(b) := {a ∈ A; a b}. It is easy to show that the following are equivalent: (1) μ is locally free; (2) every entry of P2 is 2 or larger; (3) for any a, b ∈ A, # (F(a) ∩ P(b)) ≥ 2. Recall that  is the dual group of A. For any χ ∈  and ν ∈ M(A), let 〈χ, ν〉 := ∑ a∈A χ(a) · ν{a}. It is easy to check the following. LEMMA 1.2. Let p be prime and A = Z/p. If μ is a locally free MRF on AM, then there is some c < 1 such that, for all non-trivial χ ∈  and any b∈ AS, |〈χ,μ 0 〉| ≤ c. For any χ ∈ ÂM and μ ∈ M(AM), define 〈χ , μ〉 := ∫AM χ(a) dμ[a]. A measure μ is called harmonically mixing if, for any > 0, there is some R ∈ N such that, for any χ ∈ ÂM, (rank [χ ] > R) ⇒ (|〈χ , μ〉| < ). The significance of this is the following [PY02, Theorem 12]. THEOREM A. Let A = Z/p, where p is prime. Any LCA on AM asymptotically randomizes any harmonically mixing measure. Most MRFs with full support are harmonically mixing [PY04, Theorem 15]. We now extend this. THEOREM 1.3. Let A = Z/p, where p is prime. Any locally free MRF on AM is harmonically mixing. Proof. Let μ be a locally free B-MRF. A subset I ⊂ M is B-separated if (i− j) ∈ B for all i, j ∈ I with i = j. Let K ⊂ M be finite, and let χ := ⊗k∈K χk be a character of AM. Claim 1. Let K := # (K)= rank [χ ], and let B := max {|b1 − b2|; b1,b2 ∈ B}. There exists a B-separated subset I ⊂ K such that # (I) = I ≥ K BD . (2) Proof. Let B̃ := [0 . . . B) be a box of sidelength B. Cover K with disjoint translated copies of B̃, so that K ⊂ ⊔ i∈I (B̃+ i) for some set I ⊂ K. Thus, |i − j| ≥ B for any i, j ∈ I with i = j, so (i − j) ∈ B. Also, # (B̃) = B , so each copy covers at most B points in K. Thus, we require at least K/B copies to cover all of K. In other words, I ≥ K/B . Randomization of sofic shifts by linear cellular automata 1181 Thus, χ = χI · χK\I, where χI(a) := ∏ i∈I χi(ai) and χK\I(a) := ∏ k∈K\I χk(ak). Let J := (∂I) ∪ (K \ I); fix b ∈ AJ, and let μ I ∈ M(AI) be the corresponding conditional probability measure. Since μ is a Markov random field, and the I coordinates are ‘isolated’ from one another by J coordinates, it follows that μ I is a product measure. In other words, for any a ∈ AI, μ (b) I [a] = ∏ i∈I μ (b) i {ai}. (3) Thus, the conditional expectation of χI is given as 〈χI, μ I 〉 = ∑ a∈AI μ (b) I [a] · (∏ i∈I χi(ai) ) = (∗) ∑ a∈AI (∏ i∈I μ (b) i {ai} · χi(ai) ) = ∏ i∈I (∑ ai∈A μ{ai} · χi(ai) ) = ∏ i∈I 〈χi, μ i 〉, where (∗) is by equation (3). Thus, 〈χ , μ(b)〉 = χK\I(b) · 〈χI, μ I 〉 = χK\I(b) · ∏ i∈I 〈χi, μ i 〉. Thus, if I = # (I), then |〈χ , μ(b)〉| = |χK\I(b)| · ∏ i∈I |〈χi, μ i 〉| ≤ 1 · c (4) where the last step follows from Lemma 1.2. However, 〈χ , μ〉 = ∑b∈AJ μ[b] · 〈χ , μ(b)〉, so