In this section, we'll set up a fairly abstract space called a sequence space that will play the role of the binary expansions in our understanding of the doubling map but will be applicable for other functions.
When we say that \(\rho\) defines a “notion of distance”, we mean that \(\rho\) satisfies the following properties:
Formally, a function \(\rho\) that satisfies the properties is called a metric and the sequence space \(\Lambda\) equipped with this metric \(\rho\) is called a metric space. The fact that \(\rho\) has these properties follows fairly easily from the fact that the absolute value function on \(\mathbb R\) has these properties. For example,
\begin{equation*} |x-z| \leq |x-y| + |y-z| \end{equation*}for all \(x\text{,}\) \(y\text{,}\) and \(z\) in \(\mathbb R\text{.}\) As a result, given
\begin{align*} c &= (c_0, c_1, c_2,\ldots ),\\ d &= (d_0, d_1, d_2,\ldots ), \text{ and}\\ e &= (e_0, e_1, e_2,\ldots ), \end{align*}we have
\begin{align*} \rho(c,e) &= \sum_{i=0}^{\infty} \frac{|c_i - e_i|}{2^i} = \sum_{i=0}^{\infty} \frac{|c_i - d_i + d_i - e_i|}{2^i} \\ &\leq \sum_{i=0}^{\infty} \frac{|c_i - d_i|}{2^i} + \sum_{i=0}^{\infty} \frac{|d_i - e_i|}{2^i} = \rho(c,d) + \rho(d,e). \end{align*}An intuitive interpretation of this notion of distance is as follows: Two sequences \(d\) and \(e\) are “close” iff an they agree on some long initial segment. More precisely, let \(n\in\mathbb N\) and suppose that \(d_i=e_i\) for \(0\leq i \leq n\) but that \(d_{n+1} \neq e_{n+1}\text{.}\) Then, \(\rho(d,e) \geq 1/2^{n+1}\) but
\begin{equation*} \rho(d,e) \leq \sum_{i=n+1}^{\infty} \frac{1}{2^i} = \frac{1}{2^n}. \end{equation*}The is a natural map on sequence space that turns out to display the essential features of chaos.
The shift map \(\sigma:\Lambda\to\Lambda\) is defined by
\begin{equation*} \sigma((d_0,d_1,d_2,\ldots)) = (d_1,d_2,d_3,\ldots). \end{equation*}We claim that the shift map is chaotic on the sequence space $\Lambda$. The proof very closely mirrors the proof of the the lemma and three claims that we proved in Subsection 2.7.2. We state the claims necessary to prove that the shift map is chaotic and leave their proofs as exercises.
Suppose that
\begin{equation*} d=(d_1, d_2, d_3, \cdots) \: \text{ and } \: e=(e_1, e_2, e_3, \cdots) \end{equation*}are elements of \(\Lambda\) that satisfy \(d_k \neq e_k\text{.}\) Then \(|d-e| \geq 1/2^k\text{.}\)
If anything, the proof of the Lemma 3 is easier than the corresponding proof for Lemma 2.7.3.
For every \(d\in \Lambda\) and for every \(\varepsilon > 0\text{,}\) there is some \(e\in \Lambda\) and an \(n\in\mathbb N\) such that \(\rho(d,e) < \varepsilon\) yet \(\rho(\sigma^n(d),\sigma^n(e)) \geq 1/2\text{.}\)
For every \(d\in\Lambda\) and for every \(\varepsilon > 0\text{,}\) there is some periodic orbit with an element \(e\) such that \(\rho(d,e)<\varepsilon\text{.}\)
There is a point \(d\in \Lambda\) with the property that, for every \(e\in\Lambda\) and for every \(\varepsilon > 0\) open interval \(I\subset H\text{,}\) there is some \(n\) such that \(\rho(e,\sigma^n(e))<\varepsilon\text{.}\)