We denote by $K$ the set of triplets $(\alpha,\beta,\gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha+\beta+\gamma=1$. If $(a,b,c)\in\mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$. We denote $\tau = \widehat{(-1)\,1\,\mathrm{i}}$. We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$. Let $(r_n)_{n\geq 1}$ be an element of $\{0,1\}^{\mathbf{N}^*}$. For each non-zero natural number $n$, we denote $\tilde{\tau}_n = \phi_{r_1}\circ\phi_{r_2}\circ\cdots\circ\phi_{r_n}(\tau)$. Show that $\bigcap_{n\geq 1}\tilde{\tau}_n$ is reduced to a single point belonging to $\tau$.
We denote by $K$ the set of triplets $(\alpha,\beta,\gamma)$ of $\mathbf{R}^3$ consisting of three non-negative real numbers such that $\alpha+\beta+\gamma=1$. If $(a,b,c)\in\mathbf{C}^3$, we denote by $\widehat{abc}$ the filled triangle defined by $\widehat{abc} = \{\alpha a + \beta b + \gamma c / (\alpha,\beta,\gamma)\in K\}$. We denote $\tau = \widehat{(-1)\,1\,\mathrm{i}}$. We denote by $\phi_0$ and $\phi_1$ the maps from $\mathbf{C}$ to $\mathbf{C}$ defined by: $\phi_0(z) = \frac{1+\mathrm{i}}{2}\bar{z} + \frac{-1+\mathrm{i}}{2}$ and $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$.
Let $(r_n)_{n\geq 1}$ be an element of $\{0,1\}^{\mathbf{N}^*}$. For each non-zero natural number $n$, we denote $\tilde{\tau}_n = \phi_{r_1}\circ\phi_{r_2}\circ\cdots\circ\phi_{r_n}(\tau)$.
Show that $\bigcap_{n\geq 1}\tilde{\tau}_n$ is reduced to a single point belonging to $\tau$.