Tent maps and the Cantor set

Section 2.10 Tent maps and the Cantor set

To this point, we've focused on functions that preserve an interval. In the logistic family \(f_{\lambda}(x)=\lambda x(1-x)\text{,}\) for example, we've only considered values of \(\lambda\) that satisfy \(0\leq\lambda\leq4\) so that \(f_{\lambda}:[0,1]\to[0,1]\text{.}\) In this section, we allow for the possibility that \(\lambda > 4\text{.}\) As we'll see, there is no longer an invariant interval, but rather an invariant set that we'll come to call a Cantor set.

Subsection 2.10.1 Tent maps

For each positive number \(a\text{,}\) we define the tent map \(T_a\) on \([0,1]\) by

\begin{equation*} T_a(x) = \begin{cases} a\,x \amp 0\leq x \leq 1/2 \\ a\,(1-x) \amp 1/2\leq x \leq 1. \end{cases} \end{equation*}

The graphs of several tent maps are shown in Figure 2.10.1. Note that when \(a < 2\text{,}\) the function \(T_a\) maps \([0,1]\) into itself and that when \(a=2\text{,}\) the function \(T_a\) maps \([0,1]\) onto itself. When \(a > 2\text{,}\) the unit interval is no longer invariant

Figure 2.10.1. The graphs of several tent maps

Of course, we are interested in iterating \(T_a\) for various values of \(a\text{.}\) Figure 2.10.2 shows cobweb plots of tent maps for several choices of \(a\) in the interval \((0,2]\text{.}\) It appears that \(T_a\) should have a single attractive fixed point at the origin when \(a < 1/2\) and that \(T_a\) should be chaotic on some interval when \(a > 1/2\text{.}\)

Figure 2.10.2. Cobweb plots for several tent maps

It's not too difficult to prove that \(T_2\) is chaotic on the unit interval.

Claim 2.10.3. Chaotic tent maps.

The doubling map \(d\) is semi-conjugate to the tent map \(T_2\) under the semi-conjugacy \(T_2\) itself. Thus, \(T_2\) is chaotic on the unit interval.

First, to be clear, we are claiming that

\begin{equation*} T_2 \circ T_2 = T_2 \circ d. \end{equation*}

Diagrammatically:

\begin{equation*} \begin{array}[c]{ccc} H&\stackrel{d}{\longrightarrow}&H\\ \Big\downarrow\scriptstyle{T_2}&&\Big\downarrow\scriptstyle{T_2}\\ I&\stackrel{T_2}{\longrightarrow}&I \end{array} \end{equation*}

One quick way to check if there's any sense to this is to simply graph both expressions; you should get something like Figure 2.10.4. You can do this using Desmos like so: https://www.desmos.com/calculator/nhmcgixtmz.

Figure 2.10.4. The common graph of \(T_2\circ T_2\) and \(T_2\circ d\)

The graph even reveals the common value we should shoot for, namely

\begin{equation*} T_2 \circ T_2(x) = T_2 \circ d(x) = \begin{cases} 4 x \amp 0 \le x \le \frac{1}{4} \\ 4 \left(\frac{1}{2}-x\right) \amp \frac{1}{4} \le x \le \frac{1}{2} \\ 4 \left(x-\frac{1}{2}\right) \amp \frac{1}{2} \le x \le \frac{3}{4} \\ 4 (1-x) \amp \frac{3}{4} \le x \le 1 \end{cases}. \end{equation*}

To do so, we simply check that the definitions work out over each subinterval in the piecewise definition. Over the third subinterval \(1/2 \le x \le 3/4\text{,}\) for example, we have

\begin{equation*} T_2 \circ d(x) = 2(2x-1) = 4x-1 \end{equation*}

and

\begin{equation*} T_2 \circ T_2(x) = 2(1-2(1-x)) = 4x-1. \end{equation*}

When checking these inequalities, simply check where the output of the inner function lies to ensure that you apply the correct formula for the outer function.

Just as with the chaotic quadratic, Claim 2.10.3 implies that the tent map \(T_2\) is chaotic. You can apply the conjugacy function \(T_2\) itself to a periodic point of \(d\) to find a period point of \(T_2\text{.}\) For example, \(T_2(0_{\dot 2}\overline{001}) = 2/7\) is a period three point for \(T_2\text{,}\) as you can easily check.

Finally, it's worth noting that we can study \(T_2\) by examining its effect on binary expansions.

Checkpoint 2.10.5. The tent map applied to binary expansions.

Show that

\begin{equation*} T_2(0_{\dot 2}b_1b_2b_3b_4\cdots) = \begin{cases} 0_{\dot 2}b_2b_3b_4\cdots \amp b_1 = 0 \\ 0_{\dot 2}(1-b_2)(1-b_3)(1-b_4)\cdots \amp b_1 = 1 \end{cases}. \end{equation*}

In other words, the map shifts the digits (like the doubling map) when the first bit is zero but shifts bits and swaps the remaining bits when the first bit is one.

Subsection 2.10.2 Escaping orbits

We now begin our investigation of the tall tent map \(T_3\text{,}\) whose graph is shown in Figure 2.10.6. We see there the critical issue that the unit interval is no longer invariant. As we see, the open interval \((1/3,2/3)\) maps to values larger than \(1\text{.}\)

As it turns out, there is a very important invariant set that leads us naturally to the study of fractal geometry. To find it, we'll continue to iterate \(T_3\) to see what other points might escape. To that end, the graphs of \(T_3^2\) and \(T_3^3\) are shown in

Figure 2.10.7. The graphs of \(T_3^2\) and \(T_3^3\)

Note that the middle third has escaped from each of the closed intervals \([0,1/3]\) and \([1/3,1]\) that remained after the first iteration, leaving us with four intervals

\begin{equation*} [0,1/9],[2/9,1/3],[2/3,7/9], \text{ and } [7/9,1]. \end{equation*}

A similar process happens after the next step and, generally, after the \(n^{\text{th}}\) step. Let us denote the set of all intervals that remain after \(n\) iterations by \(C_n\text{.}\) Then \(C_0\) is the unit interval and

\begin{equation*} C_1 = [0,1/3] \bigcup [2/3,1]. \end{equation*}

Generally, \(C_n\) consists of \(2^n\) intervals of length \(1/3^n\) and the invariant set for \(T_3\) is exactly

\begin{equation*} C = \bigcap_{n=0}^{\infty} C_n. \end{equation*}

It might not seem like there's much to \(C\) but it turns out to be a very rich set and the protypical fractal, as we'll meet in the next chapter. It's called the Cantor set.