## 2 Hierarchical Sources
### _Finitary Hierarchical Sources_
Let  
\[v_{i}\stackrel{{\Delta}}{{=}}\left(\log_{2}\binom{k}{i+1},\log_{2} \binom{k}{i}\right)\]5  
By the induction hypothesis,  
\[v_{i}\stackrel{{\Delta}}{{=}}\left(\log_{2}\binom{k}{i+1},\log_{2} \binom{k}{i}\right)\]6Adding,  
\[v_{i}\stackrel{{\Delta}}{{=}}\left(\log_{2}\binom{k}{i+1},\log_{2} \binom{k}{i}\right)\]7  
By Lemma 2,  
\[v_{i}\stackrel{{\Delta}}{{=}}\left(\log_{2}\binom{k}{i+1},\log_{2} \binom{k}{i}\right)\]8  
Appealing to Equation (3), we then have  
\[v_{i}\stackrel{{\Delta}}{{=}}\left(\log_{2}\binom{k}{i+1},\log_{2} \binom{k}{i}\right)\]9  
where \(N\) is the number of permutations of the \(k\)-tuple \((\lambda_{1}M_{i},\cdots,\lambda_{k}M_{i})\). Similarly,  
\[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(T_{i}^{*}(p),k^{-1}y+k^{-1}p \cdot v_{i}),\ \ (p,y)\in\Omega_{2}\]0  
where \(N_{2}\) is the number of permutations of the \(k\)-tuple \((\lambda_{1},\cdots,\lambda_{k})\). Since \(M_{i}\) is nonsingular (its determinant is \(k\)), we have \(N=N_{2}\). Substituting the right hand sides of the previous two equations into Equation (14), we obtain Equation (13) for \(\lambda=\lambda^{*}\), completing the proof by induction. Dividing both sides of Equation (13) by \(\|\lambda\|\), and using the fact that \(\|\lambda M_{i}\|=k\|\lambda\|\), we see that  
\[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(T_{i}^{*}(p),k^{-1}y+k^{-1}p \cdot v_{i}),\ \ (p,y)\in\Omega_{2}\]1  
which by Equation (6) becomes  
\[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(T_{i}^{*}(p),k^{-1}y+k^{-1}p \cdot v_{i}),\ \ (p,y)\in\Omega_{2}\]2  
It is easy to see that  
\[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(T_{i}^{*}(p),k^{-1}y+k^{-1}p \cdot v_{i}),\ \ (p,y)\in\Omega_{2}\]3  
Therefore,  
\[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(T_{i}^{*}(p),k^{-1}y+k^{-1}p \cdot v_{i}),\ \ (p,y)\in\Omega_{2}\]4  
Equation (12) then follows since the set \(\mathbb{P}_{2}^{*}=\{p_{\lambda}:\lambda\in\Lambda(2,k)\}\) is dense in \(\mathbb{P}_{2}\) and \(h_{2,k}\) is a continuous function on \(\mathbb{P}_{2}\), completing the derivation of part(c) of Theorem 2. All that remains is to prove part(b) of Theorem 2. Let \(G=\{(p,h_{2,k}(p)):p\in\mathbb{P}_{2}\}\) be the graph of \(h_{2,k}\). Part(c) is equivalent to the property that  
\[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(T_{i}^{*}(p),k^{-1}y+k^{-1}p \cdot v_{i}),\ \ (p,y)\in\Omega_{2}\]5  
This property, together with the fact that \(\{T_{i}^{*}:i=0,1,\cdots,k-1\}\) is an IFS on \(\mathbb{P}_{2}\) with attractor \(\mathbb{P}_{2}\), allows us to conclude that \(G\) is the attractor of the IFS \(\{T_{0},\cdots,T_{i-1}\}\) (Lemma B.1 of Appendix B), and \(h_{2,k}\) is self-affine because the \(T_{i}\)'s are affine. Theorem 2(b) is therefore true. **Generating Hierarchical Entropy Function Plots. ** For each \(k\geq 2\), let \(h_{2,k}^{*}:[0,1]\rightarrow\mathbb{R}\) be the function  
\[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(T_{i}^{*}(p),k^{-1}y+k^{-1}p \cdot v_{i}),\ \ (p,y)\in\Omega_{2}\]6We can obtain \(k^{n}\) points on the plot of \(h_{2,k}^{*}\) as follows. Let \(\{T_{i}:i=0,1,\cdots,k-1\}\) be the IFS on \(\Omega_{2}\) given in Theorem 2, such that the attractor of this IFS is the graph of \(h_{2,k}\). Let \(S_{0}(k)=\{(0,1,0)\}\), and generate subsets \(S_{1}(k),S_{2}(k),\cdots,S_{n}(k)\) of \(\mathbb{R}^{3}\) by the recursion  
\[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(T_{i}^{*}(p),k^{-1}y+k^{-1}p \cdot v_{i}),\ \ (p,y)\in\Omega_{2}\]7  
Then \(S_{n}(k)\) consists of \(k^{n}\) points of the form \((x,1-x,h_{2,k}(x,1-x))\). Projecting according to  
\[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(T_{i}^{*}(p),k^{-1}y+k^{-1}p \cdot v_{i}),\ \ (p,y)\in\Omega_{2}\]8  
we obtain \(k^{n}\) points on the plot of \(h_{2,k}^{*}\). Using a Dell Latitude D620 laptop, we did \(S_{n}(k)\) computations to obtain the plots in Figure 2, as follows. * The plot of \(h_{2,2}^{*}\) used the set \(S_{24}(2)\), consisting of \(2^{24}=16777216\) points, computed in 4.2 seconds. * The plot of \(h_{2,3}^{*}\) used the set \(S_{15}(3)\) consisting of \(3^{15}=14348907\) points, computed in 3.3 seconds.