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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?

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Not clear what you mean by "naturally," since the Cantor set is an uncountably infinite set, which means it may or may not occur in "nature." It can be thought of as being a "fractal," and "fractal" structures can occur in nature, although rarely in a pure form. – Thomas Andrews Dec 27 '12 at 13:55
To add to TA's nice comment, you can find items such as the study of rainfall and earthquakes as well as other areas employing fractals. Regards – Amzoti Dec 27 '12 at 14:07
With natural I meant occurring in areas of mathematics in any other way than by construction it specifically. – malin Dec 28 '12 at 20:21
up vote 30 down vote accepted

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!

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Most frogs confine themselves to a well, but a few of them roam the vast ocean of truth and knowledge. Great answer! – Haskell Curry Dec 27 '12 at 15:09
Very nice answer. Two pages readers might find interesting: kneading theory on scholarpedia and Ornstein's amazing isomorphism theorem showing that the entropy is a complete invariant of the shifts in example 2. – Martin Dec 27 '12 at 15:27
Where can you find out more about Example 2? I would like to find a more detailed exploration of it. – Permian Oct 5 '14 at 16:30

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

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+1, and to add on that... any perfect, zero-dimensional, compact metric space is homeomorphic to the Cantor set. – Asaf Karagila Dec 28 '12 at 0:21

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

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

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

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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.

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I'm not sure this is what "naturally" means. Either way if you are going to bring up logic then one can talk about descriptive set theory and its applications in model theory and logic, in those areas I think that the Cantor set has more applications than just the compactness theorem. – Asaf Karagila Dec 27 '12 at 23:59
@Asaf: well, maybe you can; I don't know anything about descriptive set theory. – Qiaochu Yuan Dec 28 '12 at 0:08
Well, you are in Berkeley... Get the hint! :-) – Asaf Karagila Dec 28 '12 at 0:12

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

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