First part: Dynamical Systems
What is a dynamical system?
X compact metrizable space S mapping from X to itself which is continuous, injective, and surjective.
* 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 systems φ : X → Y continuous and surjective φS = Tφ (φ is equivariant)
Definition:
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 any integer n.
Minimality
(X , S) dynamical system Y ⊆ X closed subset of X SY = 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 orbit closure of each point is equal to X . In this case, X is called a minimal 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 dynamical system are orbit closures, but not each orbit closure is minimal.
* Exercise 4. Give an example of a dynamical system and a point of the system which does not have a minimal orbit closure.
* Exercise 5. Show that any dynamical system does have a point whose orbit closure is minimal.
* Exercise 6. Think about whether any of the dynamical systems you 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 any m ∈ Z,
{m,m + 1, · · · ,m + n} ∩ K 6= ∅.
THEOREM. The orbit closure of x0 is minimal if and only if for each open set containing x0, the set of return times of x0 to the open set is syndetic.
* Exercise 7. Find and sketch a proof of this theorem, and be prepared to explain your proof tomorrow during the exercise session.
* 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 for each x ∈ X . The dynamical system (X , S) is usually called an irrational 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 and puts them into a finite number of boxes, and if the number of balls is greater than the number of boxes, then one of the boxes must contain at least two balls. Today this is known as the Pigeonhole Principle (or, in German, das Schubfachprinzip).
* Exercise 9. Explain why the pigeonhole principle implies that for any open set containing 0, the return times to that set form a syndetic set in Z.
Monothetic Groups and Interval Exchanges
Kronecker’s theorem has two natural and interesting generalizations, which are essentially different. A compact metric group is monothetic if there exists an element (usually called θ) of the group which generates by itself a dense subgroup. The group itself is then necessarily Abelian, and it is not difficult to see that the mapping from the group to itself given by addition of θ yields a minimal system. A more recent generalization arises by observing that a rotation of the circle group can be seen (almost) as the operation of cutting the unit interval into two and exchanging the two pieces. If we allow a larger number of cuts and a suitable permutation of the pieces, a transformation arises which is called an interval exchange transformation. For such systems, it is interesting that under natural irrationality conditions on the piece lengths and a form of irreducibility of the permutation, they are also minimal systems. Much less is known concerning the structure of these minimal systems, whereas the minimal systems arising from 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 topology S : 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 alphabet A). Any subsystem (Y ,T ) of (X , S) is called a symbolic system.
* Exercise 10. Define a metric on X for its topology; why is S continuous under this metric? Is there a “canonical” such metric?
Two basic properties of symbolic systems
Definition: The compact metric space Y is totally disconnected if its only nonempty connected sets are the one-point subsets.
* Exercise 11. Spaces of symbolic systems are totally disconnected.
Definition: A dynamical system (X ,S) is expansive if there exists a real number e > 0 (an expansivity constant) such that for each x , x ′ ∈ X with x 6= x ′, there exists an integer n ∈ Z such that d(x , x ′) ≥ e.
* Exercise 12. Why is this definition independent of the choice of the metric d? * Exercise 13. Symbolic systems are expansive. * Exercise 14. If a dynamical system is totally disconnected and expansive, is it conjugate to a symbolic system?
Words and Languages Let (X ,S) be the shift dynamical system with alphabet A. Let n be any nonnegative integer, let w ∈ An, x ∈ X , and Y ⊆ X any subset 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 the same language. * Exercise 16. Explain the syndetic property in terms of languages.
Sliding Block Codes Let (X ,S) be any symbolic system, with alphabet A. Let B also be an alphabet.
Definition: A block code of length n for (X ,S) is a map φ : Ln(X )→ B.
Let (Y ,T ) be the shift dynamical system with alphabet B, and let m ∈ Z.
Definition: The sliding block code Φ with local rule φ and memory m is the mapping
Φ : X → Y
yi = φ(xi−m · · · xi−m+n−1)
.
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 ) is a sliding block code.
* Exercise 17. Sketch a proof of the Curtis-Hedlund-Lyndon Theorem.
* Exercise 18. Let A = B = {0, 1}, (X ,S) the shift dynamical system 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 the alphabet B = An to consist of all A-words of length n. Then φ(a1 · · · an) = a1 · · · an is a block code of length n. If we take the memory m = 0, we obtain a sliding block code Φ from (X , S) to a subsystem of the shift dynamical system on the alphabet B. This subsystem is called the n-block presentation of (X , S), and is conjugate to (X ,S).
* Exercise 19. Determine the inverse of Φ.
* Exercise 20. Define the n-block presentation of any symbolic system, using Φ, and show that it is conjugate to the symbolic system.
Third part: Substitution Systems
Substitutions, Morphisms, and Languages A finite alphabet c cardinality of A
Definition: A substitution on A is a mapping from A to A∗ = ∪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 are in one-to-one correspondence with morphisms of A∗. The morphism corresponding to the substitution α will also be denoted by α. Further, a substitution α generates a continuous mapping from AZ to itself, also denoted by α.
Definition: Let α be a substitution on A. The language of α is the set
Lα = {w : ∃n∃a such that w is a factor of αn(a)}
Examples of Substitutions
The Lemasurier substitution: 0→ 12 1→ 02 2→ 01
Substitution Systems and Primitivity
Let α be a substitution on A.
Definition: The substitution system of α is the symbolic subsystem (Xα,T ) of the shift dynamical system (X ,S) with alphabet 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 minimal substitution system (Xα,T ). Then, for each integer n ≥ 1, each point x ∈ Xα is an infinite concatenation of words of the form αn(a) for some a ∈ A.
Definition: α is recognizable if this infinite concatenation is unique.
* Exercise 22. Show that for recognizability, it suffices that one point is recognizable.
Example: Consider the example of a finite system above. It is not recognizable.
Theorem: If α is primitive and Xα is infinite, then α is recognizable.
The Structure System of a Constant Length Primitive Substitution
Let α be a primitive substitution of constant length L with an infinite minimal set. By recognizability, we can associate to each point x ∈ Xα, a point z(x) ∈ mathbfZ (L), the group of L-adic integers. 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 is finite to one. The dynamical system (Z(L),+1) is therefore a factor of our substitution system. It is called the structure system of the substitution α. Also for substitutions which are primitive and have infinite minimal sets, but are not of constant length, a structure system can be defined, which is more complicated. We mention one result which can be proved using the structure theorem, on the next slide.
Coalescence
Definition: A dynamical system is said to be coalescent if any semi-conjugacy from the system to itself is one-to-one, and thus a conjugacy.
* Exercise 23: Prove that a monothetic group rotation is coalescent.
Theorem: Minimal substitution systems are coalescent.
Standard Form of a Constant Length L Substitution Let A = {1, · · · , c}, and let α be a substitution on A. Let π : A→ A be a permutation of A. If we use π to rename the letters of A, we obtain a possibly new substitution απ = παπ−1, which obviously behaves in the same manner as α - only the letters have been given different names. It is useful for classification to be able to single out one of these equivalent substitutions, which we do as follows. To any substitution α of constant length L, we associate a word of length cL by concatenating its images. Thus α(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. The standard form of the substitution α is the substitution απ whose characteristic word is lexicographically minimal.
Example: The Toeplitz substitution has characteristic word 0100. Exchanging zero and one (the only permutation) yields the substitution 0→ 11, 1→ 10, which has the characteristic word 1110. Since 0100 < 1110 lexicographically, we say that the Toeplitz substitution is in standard form.
Two Open Problems
I Given two primitive substitutions, decide whether or not their systems are conjugate.
I Given one primitive substitution, list all other substitutions (in standard form) with conjugate systems.