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). a n 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: T P = { w : P ∩ J [ w ] � = ∅} . T P has no dead ends, i.e., if w ∈ T P , then either w ⌢ 0 ∈ T P or w ⌢ 1 ∈ T P . P is decidable if T P is computable. [ T ] is the set of infinite paths through T : x ∈ [ T ] ⇐ ⇒ ( ∀ n ) x ⌈ n ∈ T. P ⊆ 2 N is closed IFF P = [ T ] for some tree T . P is effectively closed (a Π 0 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 ∈ Σ N , 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 : Σ N → Σ N be a computable (hence continuous) function. Let U 0 , U 1 , . . . , U p be a partition of Σ N into clopen sets. The itinerary It ( X ) for X ∈ Σ N is defined by ⇒ F n ( x ) ∈ U i . It ( X )( n ) = i ⇐ Let IT ( F ) = { It ( X ) : X ∈ Σ N . 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 Π 0 1 subshift. FACT: Any computable image of Σ N must be a decidable Π 0 1 class.

Symbolic Dynamics on Σ N 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 ⊆ Σ N be a decidable, subsimilar Π 0 1 class which meets J [ i ] for all i . Then there exists a partition U 0 , . . . , U k of Σ N into clopen sets and a computable F on Σ N such that Q = IT [ F ].

Avoidable Words w 1 is a factor of w if w = uw 1 v for some u, v ∈ Σ ∗ ; similarly w is a factor of x ∈ Σ N if x = uwy for some y ∈ Σ N For G ⊆ Σ ∗ and X ∈ Σ N , x avoids G if no factor of X is in G . G may be thought of as a set of forbidden words. S G is the set of words X ∈ Σ N which avoid G . FACT: Q ⊆ Σ N is a subshift if and only if Q = S G for some G ⊆ Σ + . G is avoidable if S G is nonempty.

Effective Avoidance From Cenzer-Dashti-King (MLQ 2008) Lem 2. For any sequence X 0 , X 1 , . . . of elements of 2 N , there is a nonempty subshift containing no X i . SKETCH: Let l n = 3(2 n ( n +3) ). Let w n = x ⌈ 2 ℓ n , G = { w n : n ∈ N , and let P = S G Thm 3. For any sequence n 0 < n 1 . . . and any set S = { v k : k ∈ N } such that | v k | = ℓ n k , S is avoidable. If φ ( n k ) = v k is a partial computable function, then there is a nonempty Π 0 1 subshift which avoids every v k . Thm 4. There is a nonempty Π 0 1 subshift P with no computable elements (hence with T P not computable). SKETCH: Let φ ( k ) = v k = φ k ⌈ 2 ℓ k , if defined.

Degrees of Difficulty Using the methods of CDK08, we can show Thm 5. (i) There exist subsimilar Π 0 1 classes P and Q which are Medvedev incomparable. (ii)] There exist subsimilar Π 0 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 . D α +1 ( P ) = D ( D α ( P )) D λ ( P ) = � α<λ D α ( P ). The CB rank of a countable closed set P is the least ordinal α such that D α +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, σ 2 X, . . . } is finite and therefore σ n X = σ n + k X for some n and k . Thus σ n X is periodic and hence X is eventually periodic.

Decidability of Rank One Subshifts Thm 6. Let Q be a Π 0 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 = 0 ω . Let Q n = { X : 0 n 1 X ∈ Q } , so Q 0 ⊇ Q 1 ⊃ Q 2 · · · Each Q n is finite since D ( Q n ) = ∅ . There exists m and K = { B 0 , B 1 , . . . , B k } , with each B i computable, so Q n = K for n ≥ m . For i < m , Q i is a finite set of computable reals. It follows that Q is decidable. Cor. There is a Π 0 1 class of rank one which is not isomorphic to any subshift of rank one.

Rank Two Subshifts Thm 7. Let Q be a Π 0 1 subshift of rank two. Then the set of Turing degrees of members of Q is finite. Sketch: D 2 ( Q ) is finite so its elements are all eventually periodic. For simplicity suppose that D 2 ( Q ) = { 0 ω } and let Q n = { X : 0 n 1 X ∈ Q } . Then for each n , D ( Q n ) is finite and included in D ( Q 0 ), 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 Π 0 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 D ω ( Q ) is finite. For simplicity suppose that D ω ( Q ) = { 0 ω } and again let Q n = { X : 0 n 1 X ∈ Q } . For each n , D ω ( Q n ) = ∅ so by compactness, D k ( Q n ) = ∅ for some k . Let D k ( Q 0 ) = ∅ . It follows that D k ( Q n ) = ∅ for all n and hence D k +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 Π 0 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 D α +2 ( Q ) = { 0 ω } 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 H n mapping P onto Q n , 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 Π 0 1 subshift Q with D 2 ( Q ) = { 0 ω } and such that, for each n , D ( Q n ) is a singleton which is Turing equivalent to A .

Illustration Let A have degree 0 ′ and have the form 0 n 0 10 n 1 1 · · · where n 0 < n 1 < · · · represent a modulus of convergence. Then there is a Π 0 1 subshift Q such that D 2 ( Q ) = { 0 ω with elements of rank one of the form 0 n 10 n k +1 10 n k +1 · · · , where n ≤ n k and isolated elements of the form 0 n 10 n k +1 10 n k +1 1 · · · 0 n t 10 ω , where n ≤ n k Consider also A = (01001000100001 . . . ), which is not eventually periodic and hence cannot be- long to a finite subshift. Then A is isolated in the Π 0 1 subshift Q with D ( Q ) = { 0 ω } and also containing isolated paths σ n A for all n .