Where do Cantor sets naturally occur?

Question

Cantor sets in general of course have many interesting properties on their own, and are also often used as examples of sets with these properties, but do they naturally occur in any application?

Answer 1

Cantor sets appear naturally in dynamical systems all the time.

Example 1: Consider the map $f\colon \mathbb{R}\to \mathbb{R}$ given by $f(x) = 5x(1-x)$. In dynamics, we are interested in the behavior of points under iteration of the map $f$. In other words, if we start off with a given point $x_0\in \mathbb{R}$, we are interested in the sequence $\{f^{\circ n}(x_0)\}_{n \geq 1}$. It is not hard to show that if $x_0\notin [0,1]$, then $f^{\circ n}(x_0)\to -\infty$ as $n\to \infty$. This gives us a dichotomy:

1. Either $x_0$ is such that $f^{\circ m}(x_0)\notin [0,1]$ for some $m\geq 1$, in which case $f^{\circ n}(x_0)\to -\infty$, or
2. $x_0$ is such that $f^{\circ n}(x_0)\in [0,1]$ for all $n\geq 1$, i.e., the orbit of $x_0$ is bounded.

The second case is the most interesting. The set $B$ of points $x_0$ with bounded orbits is exactly $B = \bigcap_{n\geq 1} f^{-n}([0,1])$, which is a Cantor set. Moreover, the dynamics of $f$ on $B$ is easily described (see the next example).

Example 2: Let $A$ be a finite set, and let $S = A^{\mathbb{N}}$ be the set of all infinite sequences of elements of $A$. An element $s\in S$ is then $s = (s_0,s_1,s_2,\ldots)$ where $s_i\in A$ for each $i$. Define a map $\sigma\colon S\to S$ obtained by shifting the sequence once place to the left: $$\sigma(s_0,s_1,s_2,\ldots) = (s_1,s_2,s_3,\ldots).$$ The dynamics of $\sigma$ on $S$ models many interesting dynamical systems that appear in practice, which is incredibly useful, since the dynamics of $\sigma$ is so easy to understand. Moreover, if we equip $A$ with the discrete topology and $S$ with the product topology, then $S$ is a Cantor set! As an example, the dynamics of $f$ on the set $B$ in example 1 is isomorphic in a suitable sense to the dynamics of the left-shift map $\sigma$ on the space $S = \{0,1\}^\mathbb{N}$ of binary sequences.

Example 3: Let $f\colon \mathbb{C}\to \mathbb{C}$ be the map $f(z) = z^2 + c$, where $c$ is a given complex number. If $c$ lies outside the Mandelbrot Set, then the Julia Set of $f$, i.e., the set where the dynamics of $f$ is the most chaotic and interesting, is a Cantor set.

These are just a few examples in dynamics, but there are many more. I'd be interested in seeing more examples outside dynamics!

Answer 2

Note that the ring of $p$-adic integers $\mathbb Z_p$ is homeomorphic to a Cantor set.

Answer 3

The Cantor set has been experimentally observed with X-ray diffraction in connection with the Quantum Hall effect,

http://www.eng.yale.edu/reedlab/publications/24.pdf

I don't know the physical reasons behind this, you will probably have to read the literature to find out.

Answer 4

Cantor sets naturally appear in logic. For example, if you have countably many propositional variables $x_1, x_2, ...$, then the set of possible truth-assignments to them has a natural topology which can be identified with the product topology on $\{ 0, 1 \}^{\mathbb{N}}$, which is homeomorphic to the Cantor set. This topology is compact, which is roughly speaking the topological meaning behind the compactness theorem.

Answer 5

I recently came across this bit of history concerning Mandelbrot's observation that a certain noisy signal could be thought of in terms of the Cantor Set.

"... But Mandelbrot's work eventually showed that the noise was both consistent and erratic, some kind of inescapable natural feature of the system that did not disappear with increased signal strength. But more remarkably he also showed that every burst of noise also contained within it bursts of clear signal (a situation he conceived of in terms of the Cantor set). Stranger still, he found that the ratio of periods of noise to periods of clean transmission remained constant, regardless of the scale of time used to plot the phenomenon (i.e. months, days, seconds)."

From: Benoit Mandelbrot

Answer 6

The rings of Saturn have a Cantor set-like pattern.