Effective Symbolic Dynamics Douglas Cenzer and S. Ali Dashti - - PDF document

Effective Symbolic Dynamics

Douglas Cenzer and S. Ali Dashti Department of Mathematics, University of Florida

Goals

Investigate effective versions of

• 1. Subshifts of {0, 1, . . . , k}N
• 2. Countable Subshifts in particular
• 3. Symbolic dynamics of continuous functions.

Formal Languages

Let Σ be a finite alphabet (usually {0, 1, . . . , k}. A finite string w from Σ is a word. Σ∗ is the set of all finite words on Σ; Σ+ |w| denotes the length of w. w⌈m = (w(0), . . . , w(m − 1). an is the constant word aa . . . a of length n. ε is the empty word. u⌢v or just uv denotes concatenation. If w = uv, then u is a prefix of w (u ⊑ w) and v is a suffix of w. L ⊆ Σ∗ is a formal language.

Closed Sets and Trees

T ⊆ Σ∗ is a tree if closed under prefix. [w] denotes {x ∈ {0, 1, . . . , k}N : w ⊑ x}. A closed set P is identified with a tree: TP = {w : P ∩ J[w] = ∅}. TP has no dead ends, i.e., if w ∈ TP, then either w⌢0 ∈ TP or w⌢1 ∈ TP. P is decidable if TP is computable. [T] is the set of infinite paths through T: x ∈ [T] ⇐ ⇒ (∀n)x⌈n ∈ T. P ⊆ 2N is closed IFF P = [T] for some tree T. P is effectively closed (a Π0

1 class) IFF P = [T]

for some computable T.

Subshifts

The shift function on Σ∗ is defined by σ(w) = (w(1), w(2), . . . , w(|w| − 1)); for X ∈ ΣN, σ(x) = (x(1), x(2), . . . ). A tree T ⊆ Σ∗ is subsimilar, or a subshift if w ∈ T implies σ(w) ∈ T equivalently, if T is closed under suffix. A closed set Q is subsimilar, or a subshift if Q = [T] for a subsimilar tree T; equivalently, if X ∈ Q implies σ(X) ∈ Q.

Itineraries

Let F : ΣN → ΣN be a computable (hence continuous) function. Let U0, U1, . . . , Up be a partition of ΣN into clopen sets. The itinerary It(X) for X ∈ ΣN is defined by It(X)(n) = i ⇐ ⇒ F n(x) ∈ Ui. Let IT(F) = {It(X) : X ∈ ΣN. FACT: IT(F) is a subshift, since σ(It(X)) = IT(F(x)). Lem 1. The function It is computable, so that IT[X] is computable if X is computable. Thm 1. IT[F] is is a decidable Π0

1 subshift.

FACT: Any computable image of ΣN must be a decidable Π0

1 class.

Symbolic Dynamics on ΣN

The Symbolic Dynamics of a computable map F refers to the set of itineraries of F. Here is a converse to Theorem 1. Thm 2. Let Σ = {0, 1, . . . , k} be a finite alphabet and let Q ⊆ ΣN be a decidable, subsimilar Π0

1 class which meets J[i] for all i.

Then there exists a partition U0, . . . , Uk of ΣN into clopen sets and a computable F on ΣN such that Q = IT[F].

Avoidable Words

w1 is a factor of w if w = uw1v for some u, v ∈ Σ∗; similarly w is a factor of x ∈ ΣN if x = uwy for some y ∈ ΣN For G ⊆ Σ∗ and X ∈ ΣN, x avoids G if no factor of X is in G. G may be thought of as a set of forbidden words. SG is the set of words X ∈ ΣN which avoid G. FACT: Q ⊆ ΣN is a subshift if and only if Q = SG for some G ⊆ Σ+. G is avoidable if SG is nonempty.

Effective Avoidance

From Cenzer-Dashti-King (MLQ 2008) Lem 2. For any sequence X0, X1, . . . of elements of 2N, there is a nonempty subshift containing no Xi. SKETCH: Let ln = 3(2n(n+3)). Let wn = x⌈2ℓn, G = {wn : n ∈ N, and let P = SG Thm 3. For any sequence n0 < n1 . . . and any set S = {vk : k ∈ N} such that |vk| = ℓnk, S is

• avoidable. If φ(nk) = vk is a partial computable

function, then there is a nonempty Π0

1 subshift

which avoids every vk. Thm 4. There is a nonempty Π0

1subshift P

with no computable elements (hence with TP not computable). SKETCH: Let φ(k) = vk = φk⌈2ℓk, if defined.

Degrees of Difficulty

Using the methods of CDK08, we can show Thm 5. (i) There exist subsimilar Π0

1 classes

P and Q which are Medvedev incomparable. (ii)] There exist subsimilar Π0

1 classes P and

Q such that P ≤M Q. Here P ≤M Q if there is a computable (continuous) mapping F from Q into P, so that for any element X of Q, there is an element F(X) of P which is Turing reducible to X. These results have been superceded by recent work of Joe Miller.

The Cantor-Bendixson Derivative

D(P) is the set of nonisolated points of P. Dα+1(P) = D(Dα(P)) Dλ(P) =

α<λ Dα(P).

The CB rank of a countable closed set P is the least ordinal α such that Dα+1(P) = ∅. Lem 3. For any closed set P, Dσ(P) = σD(P). Lem 4. Suppose that the subshift P is finite. Then every element of P is eventually periodic. Sketch: Let X ∈ P. Then {X, σX, σ2X, . . . } is finite and therefore σnX = σn+kX for some n and k. Thus σnX is periodic and hence X is eventually periodic.

Decidability of Rank One Subshifts

Thm 6. Let Q be a Π0

1 subshift of rank one.

Then Q is decidable and every element of Q is computable. Sketch: Every element of D(Q) is eventually periodic, hence computable. The remaining elements of Q are isolated and hence com- putable. Suppose for simplicity that D(Q) = {A}. Then σA = A, so, without loss of generality, A = 0ω. Let Qn = {X : 0n1X ∈ Q}, so Q0 ⊇ Q1 ⊃ Q2 · · · Each Qn is finite since D(Qn) = ∅. There exists m and K = {B0, B1, . . . , Bk}, with each Bi computable, so Qn = K for n ≥ m. For i < m, Qi is a finite set of computable reals. It follows that Q is decidable.

• Cor. There is a Π0

1 class of rank one which is

not isomorphic to any subshift of rank one.

Rank Two Subshifts

Thm 7. Let Q be a Π0

1 subshift of rank two.

Then the set of Turing degrees of members of Q is finite. Sketch: D2(Q) is finite so its elements are all eventually periodic. For simplicity suppose that D2(Q) = {0ω} and let Qn = {X : 0n1X ∈ Q}. Then for each n, D(Qn) is finite and included in D(Q0), so that there are only finitely many elements of rank exactly one in Q. Elements of rank 0 and rank 2 are computable.

• Cor. There is a Π0

1 class of rank two which is

not isomorphic to any subshift of rank two.

Subshifts of Rank ω

Thm 8. There is no subshift of rank ω. Sketch: Suppose by way of contradiction that Q has rank ω. Then Dω(Q) is finite. For simplicity suppose that Dω(Q) = {0ω} and again let Qn = {X : 0n1X ∈ Q}. For each n, Dω(Qn) = ∅ so by compactness, Dk(Qn) = ∅ for some k. Let Dk(Q0) = ∅. It follows that Dk(Qn) = ∅ for all n and hence Dk+1(Q) = ∅, the desired contradiction.

Inverting the Derivative

Thm 9 (CCSSW86). For any real B and any tree S ≤T B′′, there is a tree T ≤T B and a homeomorphism H from [S] onto D([T]) such that X ≤T H(X) ≤T X ⊕ B′ for all X ∈ [S]. Cor (CCSSW86). For any real A with 0′ ≤T A ≤ 0′′, there is a Π0

1 class P with D(P) a

singleton which is Turing equivalent to A.

Inverting the Derivative, II

Thm 10. Let P = [S], where S ≤T B′′, with rank α. Then there is a subshift Q = [T], with T ≤T B, such that Dα+2(Q) = {0ω} and an embedding H of P into D(Q) such that X ≤T H(X) ≤T X ⊕ B′ for all X ∈ P. Furthermore, there is a uniformly continuous function Hn mapping P onto Qn, again such that X ≤T H(X) ≤T X ⊕ B′ for all X ∈ P.

• Cor. For any real A with 0′ ≤T A ≤ 0′′, there is

a Π0

1 subshift Q with D2(Q) = {0ω} and such

that, for each n, D(Qn) is a singleton which is Turing equivalent to A.

Illustration

Let A have degree 0′ and have the form 0n010n11 · · · where n0 < n1 < · · · represent a modulus of convergence. Then there is a Π0

1 subshift Q such that D2(Q) =

{0ω with elements of rank one of the form 0n10nk+110nk+1 · · · , where n ≤ nk and isolated elements of the form 0n10nk+110nk+11 · · · 0n

t 10ω, where n ≤ nk

Consider also A = (01001000100001 . . . ), which is not eventually periodic and hence cannot be- long to a finite subshift. Then A is isolated in the Π0

1 subshift Q with

D(Q) = {0ω} and also containing isolated paths σnA for all n.