Symbolic Dynamics on Innite Traces. Minimal Shift Case and Flower Shift Case.

Symbolic Dynamics on Innite Traces.

Minimal Shift Case and Flower Shift Case.

Wit Fory±, Piotr Oprocha, Sªawomir Bakalarski.

Amiens, 8.09.2010.

Let Σ be a nite set (alphabet).

Σ^ω  the set of all innite words over Σ.

Σ^∞= Σ^∗∪ Σ^ω.

A relation I ⊂ Σ × Σ is called an independence (commutation) relation if I is symmetric and irreexive.

The relation D := (Σ × Σ)\I is called a dependence relation.

Given an independence relation I we extend it to a congruence ∼_I on Σ^∗ in the following way:

u ∼_I v ⇐⇒ it is possible to transform u to v by a nite number of swaps ab → ba of independent letters.

A trace is a element of quotient space M(Σ, I ) := Σ^∗/ ∼_I.

Let t ∈ Σ^∗

|t|_a denotes the number of occurence of letter a in t.

alph(t) denotes the set of all letteres which occur in t.

A word w ∈ Σ^∗ is in Foata normal form if 1. w is an empty word or

2. there exist v1, . . . ,vn∈ Σ⁺ (Foata steps):

(a) w = v1. . .vn

(b) Each vi is a catenation of pairwise independent letters.

(c) Each vi is minimal with respect to a lexicographical ordering.

(d) For any b ∈ alph (vi+1)there exists a ∈ alph (vi)such that (a, b) ∈ D (i = 1, . . . , n − 1).

Fact: For any x ∈ Σ^∗ there exists unique w ∈ [x]∼_I in Foata normal form.

d(x, y) =(0, if x = y 2^−l(x,y)+1, if x 6= y , where

l(x, y) = number of symbols in the longest common prex of x and y.

The metric space (Σ^ω,d) is compact, which means that every sequence has a convergent subsequence.

A shift map is a map σ : Σ^ω → Σ^ω such that (σ(x))_i :=x_i+1.

A full shift over Σ is Σ^ω together with σ.

Any σ−invariant (i.e. σ(X ) ⊂ X ) and closed X ⊂ Σ^ω is called a shift (or subshift).

For w = a₁a₂. . . ∈ Σ^ω the dependence graph ϕ_G(w) = [V , E, λ]

is dened as follows:

V := N.

λ(i) := a_i for i ∈ N.

There exists an arrow (i, j) ∈ E ⇐⇒ i < j and (a_i,a_j) ∈D.

By R^ω(Σ,I ) we denote the set of all possible (up to isomorphism) dependency graphs. Let

ϕ_G : Σ^ω → R^ω(Σ,I ) be the natural projection.

Each element of R^ω(Σ,I ) is called an innite (real) trace.

Let v ∈ V and put

P(v) := {n ∈ N| ∃v1, . . . ,vn∈V , vn=v, (vi,vi+1) ∈E for i = 1, . . . , n − 1}.

We dene h(v) to be max P(v) if P(v) 6= ∅ and 1 otherwise.

For t ∈ R^ω(Σ,I ) let

F_n(t) := minimal w.r.t lexicographical ordering word w ∈ Σ^∗ consisting of all leters λ(v) such that h(v) = n.

For any t ∈ R^ω(Σ,I ) the word w = F1(t) . . . Fn(t) is in Foata normal form with Foata steps given by F_i(t).

Also t = ϕG(F1(t)F2(t) . . .).

Example

Let Σ = {a, b, c}, I = {(a, b), (b, a)}. Then we have ϕ_G(aabbbca^∞) = ϕ_G((ab)(ab)(b)(c)(a)^∞).

In R^ω(Σ,I ) we introduce metric dRsimiliary as in Σ^ω.

dR(s, t) :=(0, if s = t

2^−lR(s,t)+1, if s 6= t , where

l_R(s, t) := max{k ∈ Z| F_i(t) = F_i(s) for 1 ≤ i ≤ k}.

It is known that the space (R^ω(Σ,I ), dR) is compact.

By a full t-shift we mean the metric space (R^ω(Σ,I ), d_R) together with the map

Φ : R^ω(Σ,I ) → R^ω(Σ,I ) dened as follows

Φ(F₁(t)F₂(2)F₃(t) . . .) = ϕG(F₂(t)F₃(t) . . .).

Observe that Φ is continous.

By a t-shift we mean any Φ-invariant and closed subset of R^ω(Σ,I ).

A shift X ⊂ Σ^ω can be converted into a subset of R^ω(Σ,I ) by applying ϕG.

Question. What can be said about the structure of the set ϕ_G(X )?

We discuss this question for minimal shifts and ower shifts.

A subshift X over Σ is said to be minimal if it contains no proper, nonempty subset which is closed and σ− invariant.

Theorem

If X is a minimal subshift, then ϕ_G(X ) is closed.

Example

Let Σ = {a, b, c} and I = {(a, b), (b, a)}.

Let p = (aabbc)^∞ and put

X ={p, σ(p), . . . , σ⁴(p)} =

{(aabbc)^∞,abbcp^∞,bbcp^∞,bcp^∞,cp^∞}.

Notice that

ϕ_G((aabbc)^∞) = ϕ_G((ab)(ab)(c) . . .) ∈ ϕG(X ), but ϕG((ab)(c) . . .) /∈ ϕG(X ).

Therefore ϕ (X ) is not Φ−invariant.

Theorem (1)

Let X be a minimal shift, alph(X ) = Σ. Let Σ1, Σ₂ be a partition of Σ and assume that Σ1× Σ₂ ⊂D.

Then there exists an integer M such that the sets

Y =

M

[

i=0

Φⁱ(ϕ_G(X )) , Z = Φ^M(Y ).

are t-shifts. Furthermore Φ|_Z is minimal and ϕ_G(X ) ∩ Z 6= ∅.

π(a) =(a, a ∈ Σ\Θ

, a ∈ Θ. .

Theorem (2)

Let Θ Σ be a subalphabet of Σ and X ⊂ Σ^ωbe a minimal subshift with alph(X ) = Σ.

Put F = ((Σ × Θ) ∪ (Θ × Σ)) \ {(a, a) : a ∈ Σ}

and assume that F ⊂ I .

Then there exists a homeomorphism

η : ϕ_G(π(X )) → ϕG(X ) commuting with Φ (i.e. η ◦ Φ = Φ ◦ η).

Furthermore if F = I , then ϕ X ) and π(X ) are t-shift and shift

Two-letter alphabet.

Let X be a minimal shift over alphabet Σ = {a, b}.

If |I | = 0, then ϕG(X ) = X (up to canonical homeomorphism).

If |I | = 2, then ϕG(X ) consist of a single xed point.

Three-letter alphabet.

Theorem (3)

Let Σ = {a, b, c} and X be a minimal shift over Σ with alph(X ) = Σ. There are the following possibilities:

1 If |I | = 0, then X and ϕG(X ) are identical up to the canonical homeomorphism.

2 If |I | = 2, then there exists n > 0 such that the set

Y = Sⁿ_i=0Φⁱ(ϕ_G(X )) is the smallest (in the sense of inclusion) invariant closed set containing ϕG(X ) and Φⁿ(Y ) is minimal.

3 If |I | = 4, then ϕG(X ) is minimal and Φ|ϕ_G(X ) is conjugate with σ|_Z where Z is a minimal shift over two-letter alphabet.

4 If |I | = 6, then |ϕG(X )| = 1 (in particular it is minimal).

Four-letter alphabet

We consider only the cases when I is represented by a co-graph with four vertices.

A co-graph is either (a) A single vertex graph, or

(b) disjoint union of two co-graphs, or (c) an edge complement of a co-graph.

The familly of co-graphs with four vertices consists of the following graphs:

Let X ⊂ Σ^ω be a minimal subshift with alph(X ) = Σ. We divide the class of co-graphs into the following subclasses.

I. Graphs 1,2,3,5,6.

There exists partion Σ = Σ1∪ Σ₂such that Σ1× Σ₂⊂D. We apply theorem 1.

The set Y = S^M_i=0Φⁱ(ϕ_G(X )) is a t-shift and after a nite number of iterations it becomes a minimal subshift Z = Φ^M(Y ), where M is a positive integer which depends on the structure of X .

In this case there exists at least one letter which is independent with any other letter.

By theorem 2 the set ϕG(X ) is homeomorphic to ϕG(X⁰), where X⁰ = π(X ) is a minimal shift.

We obtain an independence relation on π(Σ) by removing the same letter.

In cases 4, 10, 9 the set ϕ_G(X ) is a minimal shift, since ϕ_G(X⁰)is a minimal t−shift.

I is as in 7. Y = ∪^M Φⁱ(ϕ (X )) is a t-shift and Z = Φ^M(Y ) is a

Example

Σ = {a, b, c, d}, I relation generated by pairs (a, b), (b, c), (c, d), (d, a).

p = (abcd)^∞, X := {p, σ(p), σ²(p), σ³(p)}.

In this case ϕG(X ) is a t−shift and we have:

ϕ_G(X ) = ϕG({(abcd)^∞, (cdab)^∞}) ∪ ϕ_G({(bcda)^∞, (dabc)^∞}) . The t-shift ϕG(X ) is a union of two periodic orbits. Thus it splits into two disjoint minimal systems.

The example shows that when I is given by graph 8, that it may happen that ϕ_G(X ) is not minimal, neither is the sum of its iterates.

Flower shifts

Let G := {w₁, . . . ,w_n} ⊂ Σ⁺ and let M = max_j=1,...,n|w_j|.

Put

X :=

M

[

k=0

σ^k({u1u2. . . : ui ∈G for all i}).

The set X together with shift function σ is said to be a ower shift generated by words w₁, . . . ,w_n.

Flower shifts are frequently referred to as renewal systems or

nitely generated systems.

Example

Let Σ = {a, b, c, d}, I = {(a, b), (b, a), (a, c), (c, a)}.

Put w₁=ad, w₂ =cb, w₃ =d and let X be the ower shift generated by w₁,w₂,w₃.

We have

ϕ_G((cb)ⁿ(ad)^∞) = ϕ_G(a(cb)ⁿd(ad)^∞) −→ ϕ_G(a(cb)^∞) /∈ ϕ_G(X ) which implies that ϕG(X ) is not closed.

For a w ∈ Σ⁺ we denote r(w) := the rightmost letter of the word w.I (a) := the set of all letters independent with a.

Theorem

Let X be a ower shift generated by words w₁, . . . ,w_n. If there exists a ∈ Σ such that:

1. w1 ∼_I au for some u ∈ Σ^∗. 2. a 6∈ alph(w₂) ⊂I (a).

3. for all j 6= 2 either

(i) alph(w_j) ⊂alph(w₂)or

(ii) r(wj) /∈alph(w2) , ({r(wj)} ×alph(w2)) ∩D 6= ∅, then ϕ_G(X ) is not closed.

Theorem

Let X be a ower shift and let w₁, . . . ,w_n be words generating X . If there exist letters a, b ∈ alph(X ) such that for any j = 1, ..., n the following conditions hold:

1. b /∈ alph(w_j) ⊂I (a) or

2. there exist u, v ∈ Σ^∗ such that (i) w_j=uav, alph(u) ⊂ I (a),

(ii) alph(v)\I (a) 6= ∅, |u|_a=0, |v|_b=0 (iii) |u|b> |v|a+1 for some j = 1..., n, then ϕ_G(X ) is not invariant.

Theorem

Let X be a ower shift generated by w1,w2 ∈ Σ⁺. If alph(w₁) ×alph(w₂) ⊂I , then ϕG(X ) is closed.

Theorem

Let X be a ower shift generated by words w₁, . . . ,w_n∈ Σ⁺ such that (r(w_i),x) ∈ D for i = 1, . . . , n and for all x ∈ Σ. Put U = {u1u2. . . |ui ∈ {w1, . . . ,wn}}. Then ϕG(X ) is closed provided that ϕG(U) is closed.

Flower shifts over two-letter alphabet. Let Σ = {a, b}

If |I | = 0, then ϕ_G(X ) = X (up to canonical homeomorphism).

Let now |I | = 2.

Proposition

Let |I | = 2. If X is a ower shift generated by words

w1,w2, . . . ,wn∈ Σ⁺, n > 0, then ϕG(X ) contains at most three

xed points and at most countable family of points eventually xed.

In particular ϕG(X ) is closed.

Proposition

Let |I | = 2 and let w₁, . . . ,w_n∈ Σ⁺. If X is a ower shift generated by words w1, . . . ,wn∈ Σ⁺, then ϕG(X ) is invariant.