WRITING AND PRESENTING MATHEMATICS PROJECT
DYNAMICS OF SOME INTERVAL MAPS AND CHAOS
INSTRUCTIONS
The various exercises in the write up should act as footholds that allow you to flesh out a narrative. Your project should be a coherent account of the mathematics and not in Exer-cise/Solution format that these notes follow.
1. PRELIMINARIES
1.1. Chaos. Let I := [0, 1] be the standard unit interval. The circle can be described as the set [0, 1]/0 ∼ 1, i.e. the interval [0, 1] where we identify the points 0 and 1 to be the same. We denote the circle by S1. On both interval and circle, we can define a distance function. In the case of the interval, this is dI
: I × I → R, dI(x, y) := |x − y|. In the case of the circle, we define dS1 : S1 × S1 → R, dS1 (x, y) := min{|x − y|, 1 − |x − y|}.
1.1.1. Exercise: Give a geometric intuition for the way the distance on S1
is defined.
The main focus of this project are interval or circle maps, i.e. maps from the interval I or the circle S1
to itself. We will consider continuous maps.
1.1.2. Exercise: Explain how the distance functions we defined can be used to talk about con-tinuity and give a definition of what it means for functions f : I → I or g : S1 → S1
to be continuous.
We need three definitions to formulate a notion of chaos for such maps. Let F : I → I be a continuous map. (The definitions for the case where F : S1 → S1 are analogous by replacing I by S1 and the distance function dI by dS1.) We will adopt the notation Fk
for the k-fold composition F ◦ F ◦ · · · F, that is the composition of F with itself k times.
Definition 1.2 (Topological Transitivity). The map F is said to be topologically transitive if for any pairs (a, b) and (c, d) of open intervals in I there exists an integer k > 0 such that the intersection Fk(a, b) ∩ (c, d) is non-empty.
Given x ∈ I and any ϵ > 0, let Nϵ(x) = {y ∈ I such that dI(x, y) < ϵ}.
Definition 1.3 (Sensitivity). The map F has sensitive dependence on initial conditions if there exists δ > 0 such that for any x ∈ I and ϵ > 0 there exists y ∈ Nϵ(x) and an integer k such that dI(F
k
(x), F
k
(y)) > δ.
For a point x ∈ I, we call the set {Fn(x) | n ∈ N} the orbit of x. The point x is said to be a periodic point if there exists an integer k > 0 such that Fk(x) = x, or equivalently the orbit of x is a finite set. The period is said to be the smallest integer k for which Fk(x) = x, or equivalently the cardinality of the orbit.
Definition 1.4. The map F is said to be chaotic on I if
(1) F is topologically transitive,
(2) F has sensitive dependence on initial conditions, and
(3) the set of periodic points of F is dense in I.
Let F : I → I be a continous map. We say that F has a dense orbit, if there exists an x ∈ I such that the set {Fn(x) | n ∈ N} is dense in I.
1.4.3. Exercise: Show that having a dense orbit implies topological transitivity.
In fact, the other implication holds as well, i.e. topological transitivity implies the existence of a dense orbit ([3]). All of this also holds for maps of S
1
.
1.5. Binary expansion. The binary expansion of a number r ∈ [0, 1] is defined by the itera-tive process
rk+1 = fractional part of 2 rk
ak+1 = integer part of 2 rk
for k ⩾ 0 and where r0 is set to be r. We write
r = 0.a1a2a3 · · ·
Similar to the usual decimal expansion, the binary expansion has the following ambiguity:
0. a1 · · · ak10 = 0. a1 · · · ak01.
Also note that 1 = 0.1.
1.5.1. Exercise: Compare binary expansion to the usual decimal expansion and give some ex-plicit examples of numbers in decimal vs. binary expansion. Discuss rational vs. irrational numbers in terms of binary expansion.
We want to see how thinking of numbers in binary expansion can be useful in order to study maps we are interested in. Consider the doubling map
D : S1 → S1
, x → 2x (mod 1).
Let σ be the right shift on the digits of the binary expansion i.e.,
σ(0. a1a2a3 · · ·) = 0. a2a3a4 · · ·
1.5.2. Exercise: Show that the doubling map D is well-defined and that it acts as the right shift σ when we represent numbers by their binary expansion.
We will see later on how describing D in this way will be useful to obtain information about orbits of elements under D.
2. PROJECT A
Let F : [0, 1] → [0, 1] be continuous. The first part of this project is devoted to proving the following theorem.
Theorem 2.1. If F has a dense orbit and its set of periodic points is dense, then F has sensitive dependence on initial conditions.
Proof. This is a sketch of a proof and it is an exercise to fill in the details. It will be useful to draw a schematic picture of the sets and points occuring in the proof in order to follow the arguments better.
We will use the following terminology: For a point x ∈ [0, 1], we call any open interval (a, b) containing x a neighbourhood of x. Furthermore, for a subset A ⊂ [0, 1], we say that the diameter of A (denoted diam(A)) is the supremum of the distance of any two points in A.
Let p ∈ [0, 1] be a periodic point of F. Let q ∈ [0, 1] be a point not in the orbit of p whose distance to the orbit of p is d > 0. Assume F does not have sensitive dependence on ini-tial conditions. Show the existence of a point x ∈ [0, 1] and a neighbourhood N(x) of x such that diam(Fn)(N(x)) < d/4 for all n ∈ N.
Let y ∈ N(x) be a periodic point. Why does such a y exist? Let Y ∈ N be the period of y.
Show that there exists a neighbourhood N(p) of p such that |Fn(z) − Fn(p)| < d/4 for all n = 0, ...,Y − 1.
Finally, let N(q) be a neighbourhood of q with diameter less than d/4.
Use that F is topologically transitive (why?) to show the existence of x′
, x
′′ ∈ N(x) and m′
, m′′ ∈ N such that Fm′
(x
′
) ∈ N(p) and Fm′′(x
′′) ∈ N(q).
Use the triangle inequality to show that |Fm′+n
(y) − Fn(p)| < d 2 for all n = 0, ...,Y − 1 and |Fm′′(y) − q| < d/2 .
Since the point y is Y-periodic, it holds that there is an n′ ∈ {0, ...,Y −1} such that Fm′+n
′
(y) = Fm′′(y).
Use the above to show that |Fn′
(p) − q| < d. Conclude that this is a contradiction and conclude the proof. □
2.1.1. Exercise: Argue in few words that the proof can be used almost analogously in the S1 case.
2.1.2. Exercise: Use the above theorem as well as the discussion in the preliminaries to give an equivalent definition of a chaotic map.
Let D : S1 → S1
, x → 2x ( mod 1) be the doubling map and recall that it can be thought of as the right shift σ on binary sequences. In the rest of this project, we want to compute the number of periodic points. Let k ∈ N and consider the set Pk
:= {x ∈ S
1
| Dk(x) = x}. Note that this is not exactly the set of periodic points of period k, but the set of periodic points of period an integer that divides k.
2.1.3. Exercise: Compute the cardinality of Pk
.
2.1.4. Exercise: Compute the number of period points of period k for the doubling map.
3. PROJECT B
We start by studying the doubling map D : S1 → S1
, x → 2x ( mod 1).
3.0.1. Exercise: Just by using the definition of the doubling map, show that D is chaotic.
Hint: You are allowed to use the theorem proved in Project A. Furthermore, it is easier to show topological transitivity directly, instead of the existence of a dense orbit. We will do the latter in the next part of the project.
Now, recall that D can be thought of as the right shift σ on binary sequences. We want to show the existence of a dense orbit. Consider the following construction: Write a list of all finite strings of 0′
s and 1′
s, i.e.
0 1 00 01 000 001 010 011 ...
and so on. Consider the number x ∈ [0, 1] which in binary expansion is given by a concate-nation of all of the above strings, i.e. x = 0.010001000001010011...
3.0.2. Exercise: Using the interpretation of the doubling map as the right shift on binary ex-pansions, show that the orbit of x under the doubling map is dense.
For the rest of this project, we want to study the so called tent map
3.0.3. Exercise: Justify the name ”tent map.”
We want to show that T is chaotic by using that we already know it for the doubling map.
Similar to the tent map, define the map The only difference to the tent map is that the domain changed from [0, 1] to S
1
.
3.0.4. Exercise: Show that h is well-defined and that T ◦ h = h ◦ D. Use this to show that T is chaotic.
REFERENCES
[1] Devaney R. An Introduction to Chaotic Dynamical Systems. Reprint of the Second Edition. Studies in Non-linearity. Westview Press, Boulder CO (2003).
[2] Sternberg, S. Dynamical Systems. Course Notes, Harvard University.
http://www.math.harvard.edu/shlomo/docs/ ˜ dynamical systems.pdf
[3] Silverman, S. On maps with dense orbits and the definition of chaos. Rocky Mountain Journal of Mathematics. Vol. 22, Number 1 (1992)