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 $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$. The map $f\in\mathcal{E}$ satisfies $Tf = f$ and $f([0,1]) = \tau$ where $\tau = \widehat{(-1)\,1\,\mathrm{i}}$. a) For $(i,j)\in\{0,1\}^2$, determine the complex expression of $\phi_i\circ\phi_j$, recognize it, specify its fixed point and the image of $\tau$. Make a drawing. b) Let $r_1, r_2, \ldots, r_p$ be elements of $\{0,1\}$. Prove that $\phi = \phi_{r_1}\circ\phi_{r_2}\circ\cdots\circ\phi_{r_p}$ has a unique fixed point which we will not necessarily try to express simply. c) Exhibit, using the map $f$, a fixed point of $\phi$. d) Show that the set $X$ of complex numbers $z$ which are fixed points of the composition of a finite number of maps $\phi_0$ and $\phi_1$ is dense in $\tau$.
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 $\phi_1(z) = \frac{1-\mathrm{i}}{2}\bar{z} + \frac{1+\mathrm{i}}{2}$. The map $f\in\mathcal{E}$ satisfies $Tf = f$ and $f([0,1]) = \tau$ where $\tau = \widehat{(-1)\,1\,\mathrm{i}}$.
a) For $(i,j)\in\{0,1\}^2$, determine the complex expression of $\phi_i\circ\phi_j$, recognize it, specify its fixed point and the image of $\tau$. Make a drawing.
b) Let $r_1, r_2, \ldots, r_p$ be elements of $\{0,1\}$. Prove that $\phi = \phi_{r_1}\circ\phi_{r_2}\circ\cdots\circ\phi_{r_p}$ has a unique fixed point which we will not necessarily try to express simply.
c) Exhibit, using the map $f$, a fixed point of $\phi$.
d) Show that the set $X$ of complex numbers $z$ which are fixed points of the composition of a finite number of maps $\phi_0$ and $\phi_1$ is dense in $\tau$.