## 4 \(h_{3,k}\) Is Self-Affine
By definition of \(\phi(\lambda M)\) in Example 4,  
\[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(pM_{i},y+p\cdot v_{i})/3,\ \ (p,y)\in\Omega_{3}\]3 \[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(pM_{i},y+p\cdot v_{i})/3,\ \ (p,y)\in\Omega_{3}\]4 \[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(pM_{i},y+p\cdot v_{i})/3,\ \ (p,y)\in\Omega_{3}\]5In view of the fact that Equations (30-32) also hold, Property (a.2) then follows if the equations  
\[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(pM_{i},y+p\cdot v_{i})/3,\ \ (p,y)\in\Omega_{3}\]6 \[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(pM_{i},y+p\cdot v_{i})/3,\ \ (p,y)\in\Omega_{3}\]7 \[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(pM_{i},y+p\cdot v_{i})/3,\ \ (p,y)\in\Omega_{3}\]8  
are valid. These equations can be seen to hold using the fact from Equation (33) that  
\[T_{i}(p,y)\stackrel{{\Delta}}{{=}}(pM_{i},y+p\cdot v_{i})/3,\ \ (p,y)\in\Omega_{3}\]9  
Thus, Property (a.2) holds. _Proof of Theorem 3._ 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 (20) for a fixed \(M\in\mathcal{M}(k)\). Let \(\phi\in\Phi(3,k)\) be the function given in Example 4 and recall that \(H_{3,k}\) denotes the entropy function \(H_{\phi}\) on \(\Lambda(3,k)\). Referring to the definition of \(T_{M}^{*}\) in Equation (17) and \(T_{M}\) in Equation (18), we see that proving Equation (20) is equivalent to proving  
\[T_{M}(p,h_{3,k}(p))=(T_{M}^{*}(p),h_{3,k}(T_{M}^{*}(p))),\ \ p\in\mathbb{P}_{3}\]0  
We first show that  
\[T_{M}(p,h_{3,k}(p))=(T_{M}^{*}(p),h_{3,k}(T_{M}^{*}(p))),\ \ p\in\mathbb{P}_{3}\]1  
The proof is by induction on \(\|\lambda\|\). Equation (35) holds for \(\|\lambda\|=1\), which is the three cases \(\lambda=(1,0,0)\), \(\lambda=(0,1,0)\), \(\lambda=(0,0,1)\). Fix \(\lambda^{*}\in\Lambda(3,k)\) for which \(\|\lambda^{*}\|>1\), and for the induction hypothesis assume that Equation (35) holds when \(\|\lambda\|\) is smaller than \(\|\lambda^{*}\|\). The proof by induction is then completed by showing that Equation (35) holds for \(\lambda=\lambda^{*}\). Let \(\phi(\lambda^{*})=(\lambda_{1},\lambda_{2},\cdots,\lambda_{k}).\) By the induction hypothesis,  
\[T_{M}(p,h_{3,k}(p))=(T_{M}^{*}(p),h_{3,k}(T_{M}^{*}(p))),\ \ p\in\mathbb{P}_{3}\]2  
Adding,  
\[T_{M}(p,h_{3,k}(p))=(T_{M}^{*}(p),h_{3,k}(T_{M}^{*}(p))),\ \ p\in\mathbb{P}_{3}\]3  
By Lemma 4, \(\phi(\lambda^{*}M)\) is a permutation of \((\lambda_{1}M,\lambda_{2}M,\cdots,\lambda_{k}M),\) and so by Equation (3),  
\[T_{M}(p,h_{3,k}(p))=(T_{M}^{*}(p),h_{3,k}(T_{M}^{*}(p))),\ \ p\in\mathbb{P}_{3}\]4  
where \(N\) is the number of permutations of the \(k\)-tuple \((\lambda_{1}M,\cdots,\lambda_{k}M)\). Similarly,  
\[T_{M}(p,h_{3,k}(p))=(T_{M}^{*}(p),h_{3,k}(T_{M}^{*}(p))),\ \ p\in\mathbb{P}_{3}\]5  
where \(N_{2}\) is the number of permutations of the \(k\)-tuple \((\lambda_{1},\cdots,\lambda_{k})\). Since \(M\) is nonsingular (its determinant is \(k\)), we must have \(N=N_{2}\). Substituting the right hand sides of the previous two equationsinto Equation (36), we obtain Equation (35) for \(\lambda=\lambda^{*}\), completing the proof by induction. Dividing both sides of Equation (35) by \(\|\lambda\|\), and using the fact that \(\|\lambda M\|=k\|\lambda\|\), we see that  
\[T_{M}(p,h_{3,k}(p))=(T_{M}^{*}(p),h_{3,k}(T_{M}^{*}(p))),\ \ p\in\mathbb{P}_{3}\]6  
which, using Equation (8), becomes  
\[T_{M}(p,h_{3,k}(p))=(T_{M}^{*}(p),h_{3,k}(T_{M}^{*}(p))),\ \ p\in\mathbb{P}_{3}\]7  
It is easy to see that  
\[T_{M}(p,h_{3,k}(p))=(T_{M}^{*}(p),h_{3,k}(T_{M}^{*}(p))),\ \ p\in\mathbb{P}_{3}\]8  
Therefore,  
\[T_{M}(p,h_{3,k}(p))=(T_{M}^{*}(p),h_{3,k}(T_{M}^{*}(p))),\ \ p\in\mathbb{P}_{3}\]9  
Equation (34) then follows since the set \(\mathbb{P}_{3}^{*}=\{p_{\lambda}:\lambda\in\Lambda(3,k)\}\) is dense in \(\mathbb{P}_{3}\) and \(h_{3,k}\) is a continuous function on \(\mathbb{P}_{3}\), completing the derivation of part(c) of Theorem 3. All that remains is to prove part(b) of Theorem 3.