## 2 Hierarchical Sources
### _Finitary Hierarchical Sources_
In case (c), we define \(\psi(r_{1},r_{2},r_{3})\) to be the \(k\times 3\) matrix whose first \(k-r_{1}\) rows are \((011)\), whose next \(k-r_{2}\) rows are \((101)\), and whose last \(k-r_{3}\) rows are \((110)\). Letting \(\phi=\psi^{*}\), we obtain finitary \(\Lambda(3,k)\)-indexed hierarchical source \(\mathcal{S}^{\phi}\). **Remarks. ** For each fixed \(k\geq 2\),  
* The source defined in Example 3 is the unique finitary \(\Lambda(2,k)\)-indexed hierarchical source. * The source defined in Example 4 is the unique finitary \(\Lambda(3,k)\)-indexed hierarchical source. This is because the matrices employed in these examples are unique up to row permutation. **Theorem 1. ** Let \(m,k\geq 2\) be arbitrary, and let \(\{S(\lambda):\lambda\in\Lambda(m,k)\}\) be any finitary \(\Lambda(m,k)\)-indexed hierarchical source. Then the source is entropy-stable and the hierarchical entropy function induced by the source can be characterized as the unique continuous function \(h:\mathbb{P}_{m}\rightarrow[0,\infty)\) such that  
\[h(\lambda/\|\lambda\|)=H(S(\lambda))/\|\lambda\|,\ \ \lambda\in\Lambda(m,k)\]  
Theorem 1 is proved in Appendix A. _Notations and Remarks._  
* Fix \(k\) to be an arbitrary integer \(\geq 2\). Let \(\{S(\lambda):\lambda\in\Lambda(2,k)\}\) be the unique finitary \(\Lambda(2,k)\)-indexed hierarchical source. \(H_{2,k}:\Lambda(2,k)\rightarrow[0,\infty)\) shall denote the entropy function \[H_{2,k}(\lambda)=H(S(\lambda)),\ \ \lambda\in\Lambda(2,k)\] For later use, we remark that \[H_{2,k}(n_{1},n_{2})=\log_{2}\left[\frac{k!}{n_{1}!n_{2}!}\right]=\log_{2} \binom{k}{n_{1}},\ \ (n_{1},n_{2})\in\Lambda_{1}(2,k)\] (5) The hierarchical entropy function induced by this source maps \(\mathbb{P}_{2}\) into \([0,\infty)\) and shall be denoted \(h_{2,k}\). The relationship between functions \(H_{2,k}\) and \(h_{2,k}\) is \[h_{2,k}(\lambda/\|\lambda\|)=H_{2,k}(\lambda)/\|\lambda\|,\ \ \lambda\in\Lambda(2,k)\] (6)* Fix \(k\) to be an arbitrary integer \(\geq 2\). Let \(\{S(\lambda):\lambda\in\Lambda(3,k)\}\) be the unique finitary \(\Lambda(3,k)\)-indexed hierarchical source. \(H_{3,k}:\Lambda(3,k)\rightarrow[0,\infty)\) shall denote the entropy function \[H_{3,k}(\lambda)=H(S(\lambda)),\ \ \lambda\in\Lambda(3,k)\] For later use, we remark that \[H_{3,k}(n_{1},n_{2},n_{3})=\log_{2}\left[\frac{k!}{n_{1}!n_{2}!n_{3}!}\right], \ \ (n_{1},n_{2},n_{3})\in\Lambda_{1}(3,k)\] (7) The hierarchical entropy function induced by this source maps \(\mathbb{P}_{3}\) into \([0,\infty)\) and shall be denoted \(h_{3,k}\). The relationship between functions \(H_{3,k}\) and \(h_{3,k}\) is \[\lambda_{1} =(2212)+(1100)=(3312)\]0 (8)  
In Section 3, we show that hierarchical entropy function \(h_{2,k}\) is self-affine for each \(k\geq 2\), and in Section 4, we show that hierarchical entropy function \(h_{3,k}\) is self-affine for each \(k\geq 2\). **3. \(h_{2,k}\) Is Self-Affine**  
An iterated function system (IFS) on a closed nonempty subset \(\Omega\) of a finite-dimensional Euclidean space is a finite nonempty set of mappings which map \(\Omega\) into itself and are each contraction mappings. Given an IFS \(\mathcal{T}\) on \(\Omega\), there exists ([6], Theorem 9.1) a unique nonempty compact set \(Q\subset\Omega\) such that  
\[\lambda_{1} =(2212)+(1100)=(3312)\]1  
\(Q\) is called the attractor of the IFS \(\mathcal{T}\). Suppose \(h:\mathbb{P}_{m}\rightarrow[0,\infty)\) is the hierarchical entropy function induced by an entropy-stable \(\Lambda(m,k)\)-indexed hierarchical source. Let \(\Omega_{m}=\mathbb{P}_{m}\times\mathbb{R}\), regarded as a metric space with the Euclidean metric that it inherits from being regarded as a closed convex subset of \(\mathbb{R}^{m+1}\). We define \(h\) to be self-affine if there is an IFS \(\mathcal{T}\) on \(\Omega_{m}\) such that  
* Each mapping in \(\mathcal{T}\) is an affine mapping. * The attractor of \(\mathcal{T}\) is \(\{(p,h(p)):p\in\mathbb{P}_{m}\}\), the graph of \(h\). For the rest of this section, \(k\geq 2\) is fixed.