## 4 \(h_{3,k}\) Is Self-Affine
** It is clear that the set of \(k^{2}\) mappings \(\{T_{M}^{*}:M\in\mathcal{M}(k)\}\) is an IFS on \(\mathbb{P}_{3}\). This fact allows one to prove (Lemma B.4 of Appendix B) that the related set of \(k^{2}\) mappings \(\{T_{M}:M\in\mathcal{M}(k)\}\) is an IFS on \(\Omega_{3}\). In the following example, we exhibit this IFS in a special case. _Example 5._ Let \(k=3\). Referring to Figure 3, we see that the \(9\) matrices in \(\mathcal{M}(3)\) are  
\[M_{1}=\left[ \begin{array}{c}300\\ 210\\ 201\end{array} \right],\ \ M_{2}=\left[ \begin{array}{c}201\\ 111\\ 102\end{array} \right],\ \ M_{3}=\left[ \begin{array}{c}210\\ 120\\ 111\end{array} \right]\]  
\[M_{4}=\left[ \begin{array}{c}102\\ 012\\ 003\end{array} \right],\ \ M_{5}=\left[ \begin{array}{c}111\\ 021\\ 012\end{array} \right],\ \ M_{6}=\left[ \begin{array}{c}120\\ 030\\ 021\end{array} \right]\]  
\[M_{7}=\left[ \begin{array}{c}111\\ 201\\ 210\end{array} \right],\ \ M_{8}=\left[ \begin{array}{c}012\\ 102\\ 111\end{array} \right],\ \ M_{9}=\left[ \begin{array}{c}021\\ 111\\ 120\end{array} \right]\]  
Following Equation (19), let \(v_{i}\in\mathbb{R}^{3}\) be the vector whose components are the \(H_{3,3}\) entropies of the rows of \(M_{i}\). Letting \(\alpha=\log_{2}3\) and \(\beta=\log_{2}6\), Formula (7) is used to obtain  
\[v_{1}=(0,\alpha,\alpha),\ \ v_{2}=(\alpha,\beta,\alpha),\ \ v_{3}=(\alpha, \alpha,\beta)\]  
\[v_{4}=(\alpha,\alpha,0),\ \ v_{5}=(\beta,\alpha,\alpha),\ \ v_{6}=(\alpha,0,\alpha)\]  
\[v_{7}=(\beta,\alpha,\alpha),\ \ v_{8}=(\alpha,\alpha,\beta),\ \ v_{9}=(\alpha, \beta,\alpha)\]  
Following Equation (18), for each \(i=1,2,\cdots,9\), let \(T_{i}:\Omega_{3}\rightarrow\Omega_{3}\) be the mapping defined by  
\[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(pM_{i},y+p\cdot v_{i})/3,\ \ (p,y)\in\Omega_{3}\]  
Theorem 3 which follows will tell us that the graph of \(h_{3,3}\) is the attractor of the IFS \(\{T_{i}\}_{i=1}^{9}\). **Theorem 3. ** Let \(k\geq 2\) be arbitrary. The following statements hold:  
* **(a):**\(\{T_{M}:M\in\mathcal{M}(k)\}\) is an IFS on \(\Omega_{3}\). * **(b):**\(h_{3,k}\) is self-affine and its graph is the attractor of the IFS in (a). * **(c):** For each \(M\in\mathcal{M}(k)\), \[T_{M}(p,h_{3,k}(p))=(T_{M}^{*}(p),h_{3,k}(T_{M}^{*}(p))),\ \ p\in\mathbb{P}_{3}\] (20)Our proof of Theorem 3 requires a couple of lemmas, which follow. **Lemma 3. ** Let \(\lambda\in\Lambda(3,k)^{+}\), let \(\phi\in\Phi(3,k)\) be the function given in Example 4, and let \(\phi(\lambda)=(\lambda_{1},\lambda_{2},\cdots,\lambda_{k})\). Suppose we write  
\[\lambda=(kq_{1}+r_{1},kq_{2}+r_{2},kq_{3}+r_{3})\]  
where the \(q_{i}\)'s are non-negative, the \(r_{i}\)'s belong to the set \(\{1,2,\cdots,k\}\), and \(r_{1}+r_{2}+r_{3}=2k\). Then  
\[\mathrm{card}\{1\leq i\leq k:\lambda_{i}=(q_{1},q_{2}+1,q_{3}+1)\} = k-r_{1} \tag{21}\] \[\mathrm{card}\{1\leq i\leq k:\lambda_{i}=(q_{1}+1,q_{2},q_{3}+1)\} = k-r_{2}\] (22) \[\mathrm{card}\{1\leq i\leq k:\lambda_{i}=(q_{1}+1,q_{2}+1,q_{3})\} = k-r_{3} \tag{23}\]  
_Proof._ If each \(r_{i}<k\), then by definition of \(\phi(\lambda)\) in Example 4, the properties Equations (21)-(23) are true. Now suppose at least one \(r_{i}=k\). Then exactly one \(r_{i}=k\) (since otherwise some \(r_{i}=0\), which is not allowed). By symmetry, we may suppose that \(r_{1}=k\). We may now express \(\lambda\) as  
\[\lambda=((q_{1}+1)k,q_{2}k+r_{2},q_{3}k+r_{3})\]  
Since \(r_{2},r_{3}\in\{1,2,\cdots,k-1\}\), and \(r_{2}+r_{3}=k\), the definition of \(\phi(\lambda)\) tells us that  
\[v_{4}=(\alpha,\alpha,0),\ \ v_{5}=(\beta,\alpha,\alpha),\ \ v_{6}=(\alpha,0,\alpha)\]0 \[v_{4}=(\alpha,\alpha,0),\ \ v_{5}=(\beta,\alpha,\alpha),\ \ v_{6}=(\alpha,0,\alpha)\]1  
Equation (24) yields Equation (23), Equation (25) yields Equation (22), and Equation (21) is vacuously true because \(k-r_{1}=0\). **Lemma 4. ** Let \(\phi\in\Phi(3,k)\) be the function given in Example 4. Properties (a.1)-(a.2) below are true for any matrix \(M\) in the set \(\mathcal{M}(k)\).