This talk on Cantor Set and Dynamical Systems was held on Friday March 31, 2017 in MC 4045. The talk was given by James Bai.


The talk will be begin on the cosnstruction of the most commonly used tenary Cantor set. The talk will then talk about the common properties of Cantor set and methods of evaluating the size of the set. Then, depending on time, a brief introduction will be given on the dynamical system and chaos.


Cantor Set

To begin with, I’ll give an example of a stereotypical cantor set. You start with an interval $[0,1]$ and cut off the middle third and repeat this process for each of the two smaller intervals.

If we let $C_0 = [0,1]$, we can describe this process mathematically as

\[ C_n = \frac{C_{n-1}}{3} \cup \Big(\frac{2}{3} + \frac{C_{n-1}}{3} \Big)\]

and then we define

\[ C = \bigcap_{n=1}^{\infty} C_n\]

$C$ is the stereotypical Cantor set.

An interesting property is that this set has measure zero, and is uncountable. Let’s define what measure zero is.

For all $\varepsilon > 0$ there exists a sequence of intervals $\{U_n\}$ such that $A \subseteq \bigcup_{n=1}^{\infty} U_n$ and that $\sum_{n =1}^{\infty} {\left|U_n\right|} < \infty$ where we naturally define the length of an interval as ${\left|(a,b)\right|} = {\left|[a,b]\right|} = b - a$.

How do we show that that Cantor set is uncountable? Well we have a surjection to the set of ternary numbers. Every number in the cantor set can represented as a ternary number without 1 in any of it’s digits (Since 1 represents the removed middle third). Look this up on Google.

In order to measure and compare the Cantor set with other fractal, we utilize something called the Hausdorff dimension.

The Hausdorff dimension is defined by first taking the Hausdorff content which is defined as

\[C_H^{d}(S) = \inf \Big\{ \sum_{i} r_i^d : r_i \text{ are balls covering S} \Big\}\]

and then we define

\[\text{dim}_H(X) = \inf\{d \geq 1 : C_H^{d} = 0\}\]

This is not easy to calculate, we are going to use a shortcut.

It is not hard to see that $C = C/3 \cup (2/3 + C/3)$ and $C/3 \cap (2/3 \cap C/3) = {\varnothing}$. A shortcut is taking the Hausdorff measure as follows,

\[ H^{t}(C) = H^{t}(C/3) + H^{t}(1/3C + 2/3) = 2H^{t}(C/3)\]
\[H^{t}(C) = 2 \cdot \frac{1}{3}^{t}H^{t}(C)\]
\[1=2 \cdot \frac{1}{3}^{t}\]


\[t = \frac{\log 2}{\log 3}\]

Thus, the cantor set has dimension $\frac{\log 2}{\log 3}$!

Note that above we assumed that $H^{t}(C)$, $H^{t}(C/3)$ and $H^{t}(1/3C + 2/3)$ is finite – this is a reasonable assumption but we will not justify why this is the case here. So take it for granted for now.

Dynamical Systems

Let’s talk about Dynamical Systems. They are systems that describe that values through time. A Chaotic dynamical system is a dynamical system which is very sensitive to initial conditions – even a small change in initial conditions results in a huge change in outcome. An example of a chaotic dynamical system is a double pendulum – look this up!

Cantor sets are related to dynamical systems! Let’s describe this.

Let’s consider the following set of functions for $u \in {{\mathbf{R}}}$

\[f_{u}(x) = u x(1 -x)\]

This equation looks nice but it is not. When $u \geq 2 + \sqrt {5}$ the above equation messes a lot of things up. Let’s define

\[\Lambda_n = \{x \in [0,1] : f_u^{n}(x) \in [0,1]\}\]

and then define

\[\Lambda = \bigcap_{n=1}^{\infty}\Lambda_n\]

This is the set of $x \in [0,1]$ such that $f_u(x)$ is in $[0,1]$ which can be repeatedly applied and it stays in $[0,1]$.

This is a Cantor set! But I forgot to give a definition of a cantor set before so let’s give the definition here.

A Cantor set is closed and bounded, perfect and has no intervals.

In order to show that that set is a Cantor set, let’s prove the following lemma.

In $\Lambda_n$, all intervals are bounded above in length by $\frac{1}{t}$ for some $t > 1$.

If x is in $\Lambda_1$, then $f^\prime(x)$ is greater than or equal to the absolute value of the derivative of $f(x)$ at

\[\frac{1}{2} \pm \frac{\sqrt{u^2 -4u}}{2u}\]

In this way we get the t which we’re looking for. The second part of the lemma can be proved using a contradiction with the first part of the lemma and the mean value theorem.

$\Lambda$ is a Cantor Set.

$\Lambda$ is clearly closed and bounded because it is an intersection of closed sets.

If you look at the Cantor middle third set, we know for sure that the endpoints of the intervals are in the Cantor set. Something similar happens in our case.

For $x \in \Lambda$ and for all $\varepsilon > 0$ we can find $n$ such that $1/t^n < \varepsilon$. And we can find two boundary points which are close to $x$ by looking at $\Lambda_n$.

And for the third property, given interval of length $\varepsilon$ we can find $n$ such that $1/t^n < \varepsilon$.


How does this relate to Chaos? Suppose we have a point $x$ and we apply $f$ to it a many times. Since $x \in \Lambda$, $f^{n}(x)$ is mapped to a point in $[0,1]$ for each $n \in {{\mathbf{N}}}$. Since $\Lambda$ contains no intervals, there is a point close to $x$ which is not in $\Lambda$. So there is a small $y$ such that $F^k(x + y) \not\in [0,1]$ for some large enough $k$. But since $\Lambda$ is perfect, there is some point very close to $x$ (but not $x$), say $x + z$ where $z$ is very small such that $f^l(z) \not \in [0,1]$ for large enough $l$.

This is an example of chaos because in our case we can move $x$ a tiny amount and get wild changes in $f$. It is a similar idea to the phrase “a Butterfly flapping its wings in New Mexico causes a hurricane in China” as even there a small change in initial conditions causes a huge change in the outcome.

In our case, since the invariant of $f$ is a cantor set, the behavior of the system is sensitive to very small changes. This can be used to prove that $f$ is topologically transitive on the set $\Lambda$ and is then used to prove that the whole function is chaotic.

For an extension on this topic, a useful source is the below publishing.

Holmgren, Richard A. A first course in discrete dynamical systems. New York: Springer, 2002. Print.