## 2 Hierarchical Sources
### _Finitary Hierarchical Sources_
Our goal is to show that the function \(h_{2,k}:\mathbb{P}_{2}\rightarrow[0,\infty)\) is self-affine, where \(h_{2,k}\) is the hierarchical entropy function induced by the unique finitary \(\Lambda(2,k)\)-indexed hierarchical source. For each \(i=0,1,\cdots,k-1\),  
* Define the matrix \[M_{i}\stackrel{{\Delta}}{{=}}\left[ \begin{array}{cc}i+1&k-i-1\\ i&k-i\end{array} \right]\]
* Define \(T_{i}^{*}:\mathbb{P}_{2}\rightarrow\mathbb{P}_{2}\) to be the mapping \[T_{i}^{*}(p)\stackrel{{\Delta}}{{=}}k^{-1}pM_{i},\ \ p\in\mathbb{P}_{2}\] (9)* Define the vector \[v_{i}\stackrel{{\Delta}}{{=}}\left(\log_{2}\binom{k}{i+1},\log_{2} \binom{k}{i}\right)\]
* Define \(T_{i}:\Omega_{2}\rightarrow\Omega_{2}\) to be the mapping \[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(T_{i}^{*}(p),k^{-1}y+k^{-1}p \cdot v_{i}),\ \ (p,y)\in\Omega_{2}\] (10) where \(p\cdot v_{i}\) denotes the usual dot product. **Remarks. ** It is clear that the set of mappings \(\{T_{i}^{*}:i=0,1,\cdots,k-1\}\) is an IFS on \(\mathbb{P}_{2}\). This fact allows one to prove (Lemma B.3 of Appendix B) that the related set of mappings \(\{T_{i}:i=0,1,\cdots,k-1\}\) is an IFS on \(\Omega_{2}\). This result is the first part of the following theorem. **Theorem 2. ** Let \(k\geq 2\) be arbitrary. The following statements hold:  
* **(a):**\(\{T_{0},T_{1},\cdots,T_{k-1}\}\) is an IFS on \(\Omega_{2}\). * **(b):**\(h_{2,k}\) is self-affine and its graph is the attractor of the IFS in (a). * **(c):** For each \(i=0,1,\cdots,k-1\), \[T_{i}(p,h_{2,k}(p))=(T_{i}^{*}(p),h_{2,k}(T_{i}^{*}(p))),\ \ p\in\mathbb{P}_{2}\] (11)  
Our proof of Theorem 2 requires the following lemma. **Lemma 2. ** Let \  
\[\mu_{s}=(q_{1}^{\prime},q_{2}^{\prime})+(0,1),\ \ r_{1}+1\leq s\leq k\]  
For \(1\leq s\leq r_{1}\), we have  
\[\lambda_{s}M_{i}=(q_{1}+1,q_{2})M_{i}=(q_{1}(i+1)+q_{2}i+i+1,q_{1}(k-i-1)+q_{2}( k-i)+k-i-1)=(q_{1}^{\prime}+1,q_{2}^{\prime})=\mu_{s}\]  
For \(r_{1}+1\leq s\leq k\), we have  
\[\lambda_{s}M_{i}=(q_{1},q_{2}+1)M_{i}=(q_{1}(i+1)+q_{2}i+i,q_{1}(k-i-1)+q_{2}( k-i)+k-i)=(q_{1}^{\prime},q_{2}^{\prime}+1)=\mu_{s}\]  
The remaining case \(r_{1}=r_{2}=0\) is much easier. We have  
\[\lambda =(q_{1}k,q_{2}k)\] \[\lambda_{s} =(q_{1},q_{2}), 1\leq s\leq k\] \[\lambda M_{i} =(q_{1}k(i+1)+q_{2}ki,q_{1}k(k-i-1)+q_{2}k(k-i))\] \[v_{i}\stackrel{{\Delta}}{{=}}\left(\log_{2}\binom{k}{i+1},\log_{2} \binom{k}{i}\right)\]0  
_Proof of Theorem 2._ We first derive part(c) and then part(b) (part(a) is already taken care of, as remarked previously). We derive part(c) by establishing Equation (11) for a fixed \(i\in\{0,1,\cdots,k-1\}\). Let \(\phi\in\Phi(2,k)\) be the function given in Example 3 and recall that \(H_{2,k}\) denotes the entropy function \(H_{\phi}\) on \(\Lambda(2,k)\). Referring to the definition of \(T_{i}^{*}\) in Equation (9) and \(T_{i}\) in Equation (10), we see that proving Equation (11) is equivalent to proving  
\[v_{i}\stackrel{{\Delta}}{{=}}\left(\log_{2}\binom{k}{i+1},\log_{2} \binom{k}{i}\right)\]1  
We first show that  
\[v_{i}\stackrel{{\Delta}}{{=}}\left(\log_{2}\binom{k}{i+1},\log_{2} \binom{k}{i}\right)\]2  
Our proof of Equation (13) is by induction on \(\|\lambda\|\). We first must verify Equation (13) for \(\|\lambda\|=1\), which is the two cases \(\lambda=(1,0)\) and \(\lambda=(0,1)\). For \(\lambda=(1,0)\), the left side of Equation (13) is the entropy of the first row of \(M_{i}\), which by Equation (5) is  
\[v_{i}\stackrel{{\Delta}}{{=}}\left(\log_{2}\binom{k}{i+1},\log_{2} \binom{k}{i}\right)\]3  
and the right side is  
\[v_{i}\stackrel{{\Delta}}{{=}}\left(\log_{2}\binom{k}{i+1},\log_{2} \binom{k}{i}\right)\]4  
Similarly, if \(\lambda=(0,1)\), both sides of Equation (13) are equal to \(\log_{2}\binom{k}{i}\). Fix \(\lambda^{*}\in\Lambda(2,k)\) for which \(\|\lambda^{*}\|>1\), and for the induction hypothesis assume that Equation (13) holds when \(\|\lambda\|\) is smaller than \(\|\lambda^{*}\|\). The proof by induction is then completed by showing that Equation (13) holds for \(\lambda=\lambda^{*}\).