Qualitative Dynamical Systems: A Tutorial

I First part: Dynamical Systems

I Second part: Symbolic Systems

I Third part: Substitution Systems

First part: Dynamical Systems

What is a dynamical system?

X compact metrizable spaceS mapping from X to itself which is continuous, injective, andsurjective.

* Exercise 1. Show that S−1 is continuous.

Definition: (X ,S) is a (discrete) dynamical system.

When are two dynamical systems the same?

(X , S) and (Y ,T ) dynamical systemsφ : X → Y continuous and surjectiveφS = Tφ (φ is equivariant)

Definition:

I φ is a semi-conjugacy

I If φ is injective, φ is a conjugacy

I (Y ,T ) is called a factor of (X , S).

* Exercise 2. Show that if φ is a conjugacy, φ−1 is a conjugacy.* Exercise 3. Sn is a conjugacy from (X , S) to itself for anyinteger n.

Minimality

(X , S) dynamical systemY ⊆ X closed subset of XSY = Y

Definition: Set T = S |Y . Then (Y ,T ) is a dynamical system.We say that it is a subsystem of (X , S).

Let x0 ∈ X . Its orbit is defined as

O(x0) = {Snx0 : n ∈ Z}

and its orbit closure by the closure O(x0). Then (O(x0),T ) is a(possibly) new dynamical system.

Definition: The dynamical system (X ,S) is minimal if the orbitclosure of each point is equal to X . In this case, X is called aminimal set and the system (X , S) a minimal system. That is,minimal systems are those having no nonempty proper subsystems.

Can we decide whether an orbit closure is minimal?

Of course, the only possible minimal subsystems of a dynamicalsystem are orbit closures, but not each orbit closure is minimal.

* Exercise 4. Give an example of a dynamical system and a pointof the system which does not have a minimal orbit closure.

* Exercise 5. Show that any dynamical system does have a pointwhose orbit closure is minimal.

* Exercise 6. Think about whether any of the dynamical systemsyou know and like are minimal or not.

Syndetic Sets

Let K ⊆ Z be a subset of the integers.

Definition: K is syndetic if there exists n ∈ N such that for anym ∈ Z,

{m,m + 1, · · · ,m + n} ∩ K 6= ∅.

THEOREM. The orbit closure of x0 is minimal if and only if foreach open set containing x0, the set of return times of x0 to theopen set is syndetic.

* Exercise 7. Find and sketch a proof of this theorem, and beprepared to explain your proof tomorrow during the exercisesession.

* Exercise 8. Prove that factors of minimal systems are minimal.

Kronecker’s Theorem

Let X = R/Z and let α ∈ X be irrational. Set S(x) = x + α foreach x ∈ X . The dynamical system (X , S) is usually called anirrational rotation.

THEOREM. (Leopold Kronecker, 1884). (X , S) is minimal.

After 140 years, the proof idea remains interesting and elementary.Kronecker reasoned that if one has a finite number of balls andputs them into a finite number of boxes, and if the number of ballsis greater than the number of boxes, then one of the boxes mustcontain at least two balls. Today this is known as the PigeonholePrinciple (or, in German, das Schubfachprinzip).

* Exercise 9. Explain why the pigeonhole principle implies that forany open set containing 0, the return times to that set form asyndetic set in Z.

Monothetic Groups and Interval Exchanges

Kronecker’s theorem has two natural and interestinggeneralizations, which are essentially different. A compact metricgroup is monothetic if there exists an element (usually called θ) ofthe group which generates by itself a dense subgroup. The groupitself is then necessarily Abelian, and it is not difficult to see thatthe mapping from the group to itself given by addition of θ yields aminimal system. A more recent generalization arises by observingthat a rotation of the circle group can be seen (almost) as theoperation of cutting the unit interval into two and exchanging thetwo pieces. If we allow a larger number of cuts and a suitablepermutation of the pieces, a transformation arises which is calledan interval exchange transformation. For such systems, it isinteresting that under natural irrationality conditions on the piecelengths and a form of irreducibility of the permutation, they arealso minimal systems. Much less is known concerning the structureof these minimal systems, whereas the minimal systems arisingfrom monothetic groups are currently well understood.

Second part: Symbolic Systems

The Shift Dynamical System

A finite set (the alphabet)X = AZ the shift space over A, a compact

metrizable space under the product topologyS : X → X the (left) shift

Sx = x ′ if for each i ∈ Z, x ′i = xi+1.

Definition: (X ,S) is the shift dynamical system (with alphabetA). Any subsystem (Y ,T ) of (X , S) is called a symbolic system.

* Exercise 10. Define a metric on X for its topology; why is Scontinuous under this metric? Is there a “canonical” such metric?

Two basic properties of symbolic systems

Definition: The compact metric space Y is totally disconnected ifits only nonempty connected sets are the one-point subsets.

* Exercise 11. Spaces of symbolic systems are totallydisconnected.

Definition: A dynamical system (X ,S) is expansive if there existsa real number e > 0 (an expansivity constant) such that for eachx , x ′ ∈ X with x 6= x ′, there exists an integer n ∈ Z such thatd(x , x ′) ≥ e.

* Exercise 12. Why is this definition independent of the choice ofthe metric d?* Exercise 13. Symbolic systems are expansive.* Exercise 14. If a dynamical system is totally disconnected andexpansive, is it conjugate to a symbolic system?

Words and LanguagesLet (X ,S) be the shift dynamical system with alphabet A. Let nbe any nonnegative integer, let w ∈ An, x ∈ X , and Y ⊆ X anysubset of X .

Definition:

I w is a word of length n.

I w is a factor of x if for some i , xi · · · xi+n−1 = w .

I Ln(x) = {w ∈ An : w is a factor of x}.I L(x) = ∪n≥0Ln(x) is the language of x .

I L(Y ) = ∪y∈YL(y) is the language of Y .

I Ln(Y ) = ∪y∈YLn(y).

* Exercise 15. Prove that if x ∈ X , its orbit closure system(O(x),T ) is minimal if and only if each point of O(x) has thesame language.* Exercise 16. Explain the syndetic property in terms oflanguages.

Sliding Block CodesLet (X ,S) be any symbolic system, with alphabet A. Let B alsobe an alphabet.

Definition: A block code of length n for (X ,S) is a mapφ : Ln(X )→ B.

Let (Y ,T ) be the shift dynamical system with alphabet B, and letm ∈ Z.

Definition: The sliding block code Φ with local rule φ andmemory m is the mapping

Φ : X → Y

given by Φ(x) = y if for each i ∈ Z,

yi = φ(xi−m · · · xi−m+n−1)

.

The Curtis-Hedlund-Lyndon Theorem

Theorem:

I Φ is a semi-conjugacy from (X , S) to (Φ(X ),T |Φ(X )).

I Any semi-conjugacy from (X , S) to a subsystem of (Y ,T ) isa sliding block code.

* Exercise 17. Sketch a proof of the Curtis-Hedlund-LyndonTheorem.

* Exercise 18. Let A = B = {0, 1}, (X ,S) the shift dynamicalsystem over A, n = 2, m = 0, and φ(ij) = i + j mod 2. .Determine Φ(X ). Is Φ a conjugacy?

Higher Block Presentations

Let A be a finite set, and let n be any positive integer. Define thealphabet B = An to consist of all A-words of length n. Thenφ(a1 · · · an) = a1 · · · an is a block code of length n. If we take thememory m = 0, we obtain a sliding block code Φ from (X , S) to asubsystem of the shift dynamical system on the alphabet B. Thissubsystem is called the n-block presentation of (X , S), and isconjugate to (X ,S).

* Exercise 19. Determine the inverse of Φ.

* Exercise 20. Define the n-block presentation of any symbolicsystem, using Φ, and show that it is conjugate to the symbolicsystem.

Third part: Substitution Systems

Substitutions, Morphisms, and LanguagesA finite alphabetc cardinality of A

Definition: A substitution on A is a mapping from A toA∗ = ∪n≥0A

n. It is of constant length L if for each a ∈ A,α(a) ∈ AL.

Notation: A∗ is a monoid under concatenation; substitutions arein one-to-one correspondence with morphisms of A∗. Themorphism corresponding to the substitution α will also be denotedby α. Further, a substitution α generates a continuous mappingfrom AZ to itself, also denoted by α.

Definition: Let α be a substitution on A. The language of α isthe set

Lα = {w : ∃n∃a such that w is a factor of αn(a)}

Examples of Substitutions

The Fibonacci substitution: 0→ 011→ 0

The Toeplitz substitution: 0→ 011→ 00

The Thue-Morse substitution: 0→ 011→ 10

The Mephisto Waltz: 0→ 0011→ 110

A finite substitution: 0→ 0101→ 101

The Lemasurier substitution: 0→ 121→ 022→ 01

Substitution Systems and Primitivity

Let α be a substitution on A.

Definition: The substitution system of α is the symbolicsubsystem (Xα,T ) of the shift dynamical system (X ,S) withalphabet A such that

Xα = {x ∈ X : L(x) ⊆ Lα}

T = S |Xα

Definition: The matrix Mα of α is the c × c matrix with entries

Mα(a, a′) = number of occurrences of a’ in α(a).

Definition: α is primitive if Mα is primitive.

* Exercise 21. If α is primitive, then (Xα,T ) is minimal.

Recognizability

Remark: Let α be a primitive substitution with associated minimalsubstitution system (Xα,T ). Then, for each integer n ≥ 1, eachpoint x ∈ Xα is an infinite concatenation of words of the formαn(a) for some a ∈ A.

Definition: α is recognizable if this infinite concatenation isunique.

* Exercise 22. Show that for recognizability, it suffices that onepoint is recognizable.

Example: Consider the example of a finite system above. It is notrecognizable.

Theorem: If α is primitive and Xα is infinite, then α isrecognizable.

The Structure System of a Constant Length PrimitiveSubstitution

Let α be a primitive substitution of constant length L with aninfinite minimal set. By recognizability, we can associate to eachpoint x ∈ Xα, a point z(x) ∈ mathbfZ (L), the group of L-adicintegers. This mapping is a semi-conjugacy, if we provide Z(L)with the mapping z → z + 1, and it is not hard to see that it isfinite to one. The dynamical system (Z(L),+1) is therefore afactor of our substitution system. It is called the structure systemof the substitution α. Also for substitutions which are primitiveand have infinite minimal sets, but are not of constant length, astructure system can be defined, which is more complicated. Wemention one result which can be proved using the structuretheorem, on the next slide.

Coalescence

Definition: A dynamical system is said to be coalescent if anysemi-conjugacy from the system to itself is one-to-one, and thus aconjugacy.

* Exercise 23: Prove that a monothetic group rotation iscoalescent.

Theorem: Minimal substitution systems are coalescent.

Standard Form of a Constant Length L SubstitutionLet A = {1, · · · , c}, and let α be a substitution on A. Letπ : A→ A be a permutation of A. If we use π to rename theletters of A, we obtain a possibly new substitution απ = παπ−1,which obviously behaves in the same manner as α - only the lettershave been given different names. It is useful for classification to beable to single out one of these equivalent substitutions, which wedo as follows. To any substitution α of constant length L, weassociate a word of length cL by concatenating its images. Thusα(1) · · ·α(c) is called the characteristic word of the substitution α.Lemma. Different substitutions have different characteristic words.Definition: Let α be a constant length L substitution. Thestandard form of the substitution α is the substitution απ whosecharacteristic word is lexicographically minimal.

Example: The Toeplitz substitution has characteristic word 0100.Exchanging zero and one (the only permutation) yields thesubstitution 0→ 11, 1→ 10, which has the characteristic word1110. Since 0100 < 1110 lexicographically, we say that theToeplitz substitution is in standard form.

Two Open Problems

I Given two primitive substitutions, decide whether or not theirsystems are conjugate.

I Given one primitive substitution, list all other substitutions (instandard form) with conjugate systems.

Next week we’ll try to explain our solutions to the secondproblem for constant length primitive substitutions.