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Section 5.2 Similarity dimension of an IFS

Equation (5.1.1) is not quite general enough to compute the fractal dimension of all self-similar sets, since iterated function systems need not have all contraction ratios equal to one another. For example, equation (5.1.1) cannot compute the dimension of the \(z\)-curve. There is an important generalization of equation (5.1.1) which defines the dimension associated with any IFS. We will assume that all iterated function systems consist of pure similarities for the remainder of this chapter.

Definition 5.2.1. Similarity dimension.

Let \(\left\{f_i\right\}_{i=1}^m\) be a fixed IFS of similarities and let \(\left\{r_i\right\}_{i=1}^m\) be the list of associated similarity ratios. Define a function \(\Phi :[0,\infty )\rightarrow \mathbb{R}\) by

\begin{equation*} \Phi (s) = r_1^s+\cdots +r_m^s. \end{equation*}

Note that \(\Phi\) is continuous, strictly decreasing, \(\Phi (0) = m,\) and \(\lim _{s\rightarrow \infty }\Phi (s) = 0.\) Thus there is a unique positive number \(s\) such that \(\Phi (s) = 1.\) This unique value of \(s\) is defined to be the similarity dimension of the IFS.

We can see that this definition agrees with that given by equation (5.1.1), when applicable. If \(r_i = r\) for each \(i=1,\ldots ,m,\) then the similarity dimension is the unique \(s\) such that

\begin{equation*} \sum _{i = 1}^m r^s = m r^s = 1, \end{equation*}

which has solution \(\frac{\log m}{\log 1/r}.\)

Suppose that \(r_1\) and \(r_2\) are positive numbers satisfying \(r_1 + r_2 \leq 1.\) We define an iterated function system \(\left\{f_1,f_2\right\}\) on \(\mathbb{R}\) by setting \(f_1(x) = r_1x\) and \(f_2(x) = r_2 x + \left(1-r_2\right).\) Note that \(f_1\) contracts the unit interval by the factor \(r_1\)towards 0, while \(f_2\) contracts the unit interval by the factor \(r_2\) towards 1. If \(r_1 + r_2 = 1\text{,}\) then the invariant set is the unit interval and the dimension is 1. If \(r_1 + r_2 < 1,\) then this IFS generalizes the Cantor construction. The dimension of this IFS is the unique solution to the equation \(r_1^s + r_2^s = 1.\) For example if \(r_1 = 1/2\) and \(r_2 = 1/4,\) we obtain

\begin{equation*} \frac{1}{2^s} + \frac{1}{4^s} = 1 \text{or} s = \frac{\log \frac{1 + \sqrt{5}}{2}}{\log 2}. \end{equation*}

This set is shown in Figure 5.2.3.

Figure 5.2.3. A two-scale Cantor set

Not all equations of this form can be solved explicitly. For example, if \(r_1 = 1/2\) and \(r_2 = 1/3\) then the similarity dimension is the unique solution to \(1\left/2^s\right. + 1\left/3^s\right. = 1.\) While this equation cannot be explicitly solved, it does uniquely characterize the dimension. Furthermore, numerical algorithms can be used to approximate the dimension to a high degree of accuracy. In this case, the dimension can be estimated by the following command.

Another example is provided by the \(z\)-curve, which has similarity ratio list \(\left\{1/3,\left.\sqrt{2}\right/6,1/3,\left.\sqrt{2}\right/6,1/3\right\}.\) It’s dimension is given by the equation

\begin{equation*} \frac{3}{3^s} + 2\left(\frac{\sqrt{2}}{6}\right)^s = 1. \end{equation*}

The solution is approximately 1.32038.

If \(r_1+ r_2 > 1,\) then the solution to \(r_1^s + r_2^s = 1\) will satisfy \(s>1.\) This indicates a potential problem with similarity dimension since we don’t want to assign number larger than one to be the dimension of a subset of \(\mathbb{R}.\)