We consider $\alpha = (\alpha_i)_{i \in I} \in (\mathbb{R}_+^*)^I$ and $\beta = (\beta_j)_{j \in J} \in (\mathbb{R}_+^*)^J$ such that $\sum_{i \in I} \alpha_i = \sum_{j \in J} \beta_j = 1$. We denote by $\boldsymbol{p}$ the element of $F(\alpha, \beta)$ defined by $p_{ij} = \alpha_i \beta_j > 0$ for all $(i,j) \in I \times J$. Let $C = (C_{ij})_{(i,j) \in I \times J} \in \mathbb{R}_+^{I \times J}$ and $\epsilon > 0$. We consider $J_\epsilon : Q \rightarrow \mathbb{R}$ defined by $$J_\epsilon(\boldsymbol{q}) = \sum_{ij} q_{ij} C_{ij} + \epsilon \operatorname{KL}(\boldsymbol{q}, \boldsymbol{p})$$ and $\boldsymbol{q}(\epsilon)$ the unique minimizer of $J_\epsilon$ on $F(\alpha, \beta)$.
(a) Verify that $q(\epsilon)_{ij} > 0$ for all $(i,j) \in I \times J$ (Hint: One may reason by contradiction and consider for all $t \in ]0,1[$ $\boldsymbol{q}(\epsilon, t) = (1-t)\boldsymbol{q}(\epsilon) + t\boldsymbol{p}$ then observe the behavior of $\varphi(x)$ near $x = 0$).
(b) Show that this is no longer true if we assume that $\epsilon = 0$.

We define a sequence of polynomials $\left( F _ { i } \right) _ { i \geqslant 0 }$ by the following recurrence formula: $$\begin{aligned} F _ { 0 } & = f _ { 0 } + f _ { 1 } X + \cdots + f _ { d } X ^ { d } \\ \text { for } i \geqslant 0 , \quad F _ { i + 1 } & = F _ { i } + R _ { i } \end{aligned}$$ where $R _ { i }$ denotes the remainder of the Euclidean division of $P$ by $F _ { i }$. We denote by $Q _ { i }$ the quotient of the Euclidean division of $P$ by $F _ { i }$. We are furthermore given $r , s \in \mathbb { R } _ { + } ^ { * }$ with $r < \rho$ and we set, for $i \in \mathbb { N }$: $$\alpha _ { i } = s ^ { - d } \cdot \left\| F _ { i } - X ^ { d } \right\| _ { r , s } ; \quad \beta _ { i } = \left\| 1 - Q _ { i } \right\| _ { r , s } ; \quad \varepsilon _ { i } = s ^ { - d } \cdot \left\| R _ { i } \right\| _ { r , s } .$$
Show that we can choose $r$ and $s$ such that $\alpha _ { 0 } + 2 \varepsilon _ { 0 } \leqslant \frac { 1 } { 3 }$ and $\beta _ { 0 } + \varepsilon _ { 0 } \leqslant \frac { 1 } { 3 }$.

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
Show that if $\varphi \in \mathcal{C}_{c}(\mathbb{R})$ is a positive function, we have $\|T_{\mu}\varphi\|_{1} = \|\varphi\|_{1}$.

We define $Q_{>0} = (\mathbb{R}_+^*)^{I \times J}$ and $\mathscr{L} : Q_{>0} \times (\mathbb{R}^I \times \mathbb{R}^J) \rightarrow \mathbb{R}$ defined by $$\mathscr{L}(\boldsymbol{q}, (f, g)) = J_\epsilon(\boldsymbol{q}) + \sum_{i \in I} f_i \left(\alpha_i - \sum_{j \in J} q_{ij}\right) + \sum_{j \in J} g_j \left(\beta_j - \sum_{i \in I} q_{ij}\right).$$ (a) Verify that $Q_{>0}$ is an open convex set of $\mathbb{R}^{I \times J}$.
(b) Show that there exists $(f(\epsilon), g(\epsilon)) \in \mathbb{R}^I \times \mathbb{R}^J$ such that $\mathscr{L}(q(\epsilon), (f(\epsilon), g(\epsilon)))$ is a saddle point of $\mathscr{L}$. (Hint: One may identify $\mathbb{R}^{I \times J}$ with $\mathbb{R}^n$ and $\mathbb{R}^I \times \mathbb{R}^J$ with $\mathbb{R}^m$, for $n$ the cardinality of $I \times J$ and $m$ the sum of the cardinalities of $I$ and $J$, then use question 3 of part I.)

We define a sequence of polynomials $\left( F _ { i } \right) _ { i \geqslant 0 }$ by the following recurrence formula: $$\begin{aligned} F _ { 0 } & = f _ { 0 } + f _ { 1 } X + \cdots + f _ { d } X ^ { d } \\ \text { for } i \geqslant 0 , \quad F _ { i + 1 } & = F _ { i } + R _ { i } \end{aligned}$$ where $R _ { i }$ denotes the remainder of the Euclidean division of $P$ by $F _ { i }$. We denote by $Q _ { i }$ the quotient of the Euclidean division of $P$ by $F _ { i }$. We set, for $i \in \mathbb { N }$: $$\alpha _ { i } = s ^ { - d } \cdot \left\| F _ { i } - X ^ { d } \right\| _ { r , s } ; \quad \beta _ { i } = \left\| 1 - Q _ { i } \right\| _ { r , s } ; \quad \varepsilon _ { i } = s ^ { - d } \cdot \left\| R _ { i } \right\| _ { r , s } .$$
Verify that, for all $i \in \mathbb { N }$, we have the relation: $$\left( 1 - Q _ { i } \right) \cdot R _ { i } = \left( Q _ { i + 1 } - Q _ { i } \right) \cdot F _ { i + 1 } + R _ { i + 1 }$$

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
Show that for all $k \geqslant 0$, if $\varphi \in \mathcal{C}_{c}^{k}(\mathbb{R})$ then $T_{\mu}\varphi \in \mathcal{C}_{c}^{k+1}(\mathbb{R})$. Also show that
$$\left\|(T_{\mu}\varphi)^{(k)}\right\|_{\infty} \leqslant \left\|\varphi^{(k)}\right\|_{\infty}$$

We define $Q_{>0} = (\mathbb{R}_+^*)^{I \times J}$ and $\mathscr{L} : Q_{>0} \times (\mathbb{R}^I \times \mathbb{R}^J) \rightarrow \mathbb{R}$ defined by $$\mathscr{L}(\boldsymbol{q}, (f, g)) = J_\epsilon(\boldsymbol{q}) + \sum_{i \in I} f_i \left(\alpha_i - \sum_{j \in J} q_{ij}\right) + \sum_{j \in J} g_j \left(\beta_j - \sum_{i \in I} q_{ij}\right).$$ (a) Show that for all $(f, g) \in \mathbb{R}^I \times \mathbb{R}^J$, the minimum of $\boldsymbol{q} \mapsto \mathscr{L}(\boldsymbol{q}, (f, g))$ on $Q_{>0}$ is attained at $q(f,g)_{ij} = e^{(f_i + g_j - C_{ij})/\epsilon} p_{ij}$.
(b) Calculate the value of $G(f, g) = \mathscr{L}(q(f,g), (f,g))$.
(c) Verify that $G$ is concave on $\mathbb{R}^I \times \mathbb{R}^J$.

We define a sequence of polynomials $\left( F _ { i } \right) _ { i \geqslant 0 }$ by the following recurrence formula: $$\begin{aligned} F _ { 0 } & = f _ { 0 } + f _ { 1 } X + \cdots + f _ { d } X ^ { d } \\ \text { for } i \geqslant 0 , \quad F _ { i + 1 } & = F _ { i } + R _ { i } \end{aligned}$$ where $R _ { i }$ denotes the remainder of the Euclidean division of $P$ by $F _ { i }$. We denote by $Q _ { i }$ the quotient of the Euclidean division of $P$ by $F _ { i }$. We set, for $i \in \mathbb { N }$: $$\alpha _ { i } = s ^ { - d } \cdot \left\| F _ { i } - X ^ { d } \right\| _ { r , s } ; \quad \beta _ { i } = \left\| 1 - Q _ { i } \right\| _ { r , s } ; \quad \varepsilon _ { i } = s ^ { - d } \cdot \left\| R _ { i } \right\| _ { r , s } .$$
Show that, for all $i \in \mathbb { N }$, we have $\alpha _ { i + 1 } \leqslant \alpha _ { i } + \varepsilon _ { i }$ and if $\alpha _ { i + 1 } < 1$ then: $$\beta _ { i + 1 } \leqslant \beta _ { i } + \frac { \beta _ { i } \varepsilon _ { i } } { 1 - \alpha _ { i + 1 } } \quad \text { and } \quad \varepsilon _ { i + 1 } \leqslant \frac { \beta _ { i } \varepsilon _ { i } } { 1 - \alpha _ { i + 1 } } .$$

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
For $k \geqslant 1$, show that if $\varphi \in \mathcal{C}_{c}^{k+1}(\mathbb{R})$, we have
$$\left\|(T_{\mu}\varphi)^{(k)} - \varphi^{(k)}\right\|_{\infty} \leqslant \frac{\mu}{2} \left\|\varphi^{(k+1)}\right\|_{\infty}.$$

Verify that if $f_* : \mathbb{R}^J \rightarrow \mathbb{R}^I$ and $g_* : \mathbb{R}^I \rightarrow \mathbb{R}^J$ are defined by $$f_*(g)_i = -\epsilon \log\left(\sum_{j \in J} e^{(g_j - C_{ij})/\epsilon} \beta_j\right) \text{ and } g_*(f)_j = -\epsilon \log\left(\sum_{i \in I} e^{(f_i - C_{ij})/\epsilon} \alpha_i\right)$$ then for all $(f, g) \in \mathbb{R}^I \times \mathbb{R}^J$, we have $\frac{\partial G}{\partial f_i}(f_*(g), g) = \frac{\partial G}{\partial g_j}(f, g_*(f)) = 0$ for all $(i,j) \in I \times J$.

We define a sequence of polynomials $\left( F _ { i } \right) _ { i \geqslant 0 }$ by the following recurrence formula: $$\begin{aligned} F _ { 0 } & = f _ { 0 } + f _ { 1 } X + \cdots + f _ { d } X ^ { d } \\ \text { for } i \geqslant 0 , \quad F _ { i + 1 } & = F _ { i } + R _ { i } \end{aligned}$$ where $R _ { i }$ denotes the remainder of the Euclidean division of $P$ by $F _ { i }$. We denote by $Q _ { i }$ the quotient of the Euclidean division of $P$ by $F _ { i }$. We set, for $i \in \mathbb { N }$: $$\alpha _ { i } = s ^ { - d } \cdot \left\| F _ { i } - X ^ { d } \right\| _ { r , s } ; \quad \beta _ { i } = \left\| 1 - Q _ { i } \right\| _ { r , s } ; \quad \varepsilon _ { i } = s ^ { - d } \cdot \left\| R _ { i } \right\| _ { r , s } .$$
Deduce that, for all $i \in \mathbb { N }$, we have:

• $\alpha _ { i } \leqslant \alpha _ { 0 } + 2 \cdot \left( 1 - 2 ^ { - i } \right) \cdot \varepsilon _ { 0 }$,
• $\beta _ { i } \leqslant \beta _ { 0 } + \left( 1 - 2 ^ { - i } \right) \cdot \varepsilon _ { 0 }$,
• $\varepsilon _ { i } \leqslant 2 ^ { - i } \cdot \varepsilon _ { 0 }$.

In this part we consider a map $\varphi$ from $\mathbf { R } ^ { n }$ to $\mathbf { R } ^ { n }$ of class $\mathcal { C } ^ { 1 }$ such that $\varphi ( 0 ) = 0$, and denoting $a = d \varphi ( 0 )$, such that all eigenvalues of $a$ have strictly negative real part.
Show that the function $$b : \left\lvert \, \begin{array} { r l l } \mathbf { R } ^ { n } \times \mathbf { R } ^ { n } & \rightarrow & \mathbf { R } \\ ( x , y ) & \mapsto & \int _ { 0 } ^ { + \infty } \left\langle e ^ { t a } ( x ) \mid e ^ { t a } ( y ) \right\rangle d t \end{array} \right.$$ is well-defined and that it defines an inner product on $\mathbf { R } ^ { n }$.

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
We assume that $(\mu_{n})_{n \geqslant 1}$ is a sequence of strictly positive real numbers such that $\sum_{n \geqslant 1} \mu_{n}$ converges. We fix $\psi_{0} \in \mathcal{C}_{c}(\mathbb{R})$ and we define by recursion the sequence $(\psi_{n})_{n \geqslant 0}$ by
$$\forall n \geqslant 0,\quad \psi_{n+1} = T_{\mu_{n+1}} \psi_{n}$$
Show that for all $n \geqslant k$, $\psi_{n}$ is of class $C^{k}$.

Let $(f^0, g^0) \in \mathbb{R}^{I \times J}$. For all $k \geq 0$, we consider $$g^{k+1} = g_*(f^k) \text{ and } f^{k+1} = f_*(g^{k+1})$$ Show that the sequence $(G(f^k, g^k))_{k \geq 0}$ is increasing.

We define a sequence of polynomials $\left( F _ { i } \right) _ { i \geqslant 0 }$ by the following recurrence formula: $$\begin{aligned} F _ { 0 } & = f _ { 0 } + f _ { 1 } X + \cdots + f _ { d } X ^ { d } \\ \text { for } i \geqslant 0 , \quad F _ { i + 1 } & = F _ { i } + R _ { i } \end{aligned}$$ where $R _ { i }$ denotes the remainder of the Euclidean division of $P$ by $F _ { i }$. We denote by $Q _ { i }$ the quotient of the Euclidean division of $P$ by $F _ { i }$.
15a. Show that the sequence $\left( F _ { i } \right) _ { i \geqslant 0 }$ converges for the norm $\| \cdot \| _ { r , s }$ towards a monic polynomial $F \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n } [ X ] \right)$ of degree $d$ which satisfies $F _ { \mid t = 0 } = X ^ { d }$.
15b. Show that there exists $G \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n } [ X ] \right)$ such that $P = F G$.

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let the application $f_A$ defined on $\mathbf{R}$ by $$f_A(t) = \operatorname{det}(A + tM).$$ Show that there exists $\varepsilon_0 > 0$ such that, for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$.

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
We assume that $(\mu_{n})_{n \geqslant 1}$ is a sequence of strictly positive real numbers such that $\sum_{n \geqslant 1} \mu_{n}$ converges. We fix $\psi_{0} \in \mathcal{C}_{c}(\mathbb{R})$ and we define by recursion the sequence $(\psi_{n})_{n \geqslant 0}$ by
$$\forall n \geqslant 0,\quad \psi_{n+1} = T_{\mu_{n+1}} \psi_{n}$$
Show that for all $k \in \mathbb{N}$ and $n \geqslant k+2$, we have
$$\left\|\psi_{n+1}^{(k)} - \psi_{n}^{(k)}\right\|_{\infty} \leqslant \frac{\mu_{n+1}}{2} \left\|\psi_{k+1}^{(k+1)}\right\|_{\infty}.$$
Deduce that for all $k \in \mathbb{N}$, the sequence of functions $\psi_{n}^{(k)}$ converges uniformly on $\mathbb{R}$.

We fix $x \in \mathcal{C}$ such that $\|Ax\| = 1$. Let $B \in \mathrm{SO}(\mathbb{R}^2)$ be a matrix such that $x = B\binom{1}{0}$. a) Show that for all $r \in ]0,1[$ there exists $x_r \in \mathcal{C}$ such that $$\left\| AB\begin{pmatrix} r & 0 \\ 0 & \frac{1}{r} \end{pmatrix} x_r \right\| > 1$$ b) Show that if $x_r = \binom{y_r}{z_r}$, then $z_r^2 > \dfrac{r^2}{1+r^2}$.

Let $(f^0, g^0) \in \mathbb{R}^{I \times J}$. For all $k \geq 0$, we consider $$g^{k+1} = g_*(f^k) \text{ and } f^{k+1} = f_*(g^{k+1})$$ Assume that there exist $f^\infty = (f_i^\infty)_{i \in I}$ and $g^\infty = (g_j^\infty)_{j \in J}$ such that $|f_i^k - f_i^\infty| \rightarrow 0$ and $|g_j^k - g_j^\infty| \rightarrow 0$ for all $i \in I$ and $j \in J$. We denote $G_* = \sup\{G(f,g) \mid (f,g) \in \mathbb{R}^I \times \mathbb{R}^J\}$.
(a) Show that $G(f^\infty, g^\infty) = G_*$.
(b) Show that $G(f(\epsilon), g(\epsilon)) = G_*$.
(c) Show that there exists a constant $a \in \mathbb{R}$ such that $f(\epsilon)_i = f_i^\infty + a$ and $g(\epsilon)_j = g_j^\infty - a$ for all $(i,j) \in I \times J$.
(d) Deduce that $q(f^k, g^k) \rightarrow q(\epsilon)$.

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
We assume that $(\mu_{n})_{n \geqslant 1}$ is a sequence of strictly positive real numbers such that $\sum_{n \geqslant 1} \mu_{n}$ converges. We fix $\psi_{0} \in \mathcal{C}_{c}(\mathbb{R})$ and we define by recursion the sequence $(\psi_{n})_{n \geqslant 0}$ by
$$\forall n \geqslant 0,\quad \psi_{n+1} = T_{\mu_{n+1}} \psi_{n}$$
Show that the limit $f = \lim_{n \rightarrow \infty} \psi_{n}$ is of class $C^{\infty}$, and that for all $k \geqslant 0$ we have
$$\left\|f^{(k)}\right\|_{\infty} \leqslant \left\|\psi_{k}^{(k)}\right\|_{\infty}.$$

Let $f \in \mathscr { D } _ { \rho } ( \mathbb { R } )$ such that $f ( 0 ) > 0$. Show that there exists $\rho _ { f } \in \mathbb { R } _ { + } ^ { * }$ such that $\rho _ { f } \leqslant \rho$ and such that $f > 0$ on $U _ { \rho _ { f } }$ and $\sqrt { f } \in \mathscr { D } _ { \rho _ { f } } ( \mathbb { R } )$.

In this part we consider a map $\varphi$ from $\mathbf { R } ^ { n }$ to $\mathbf { R } ^ { n }$ of class $\mathcal { C } ^ { 1 }$ such that $\varphi ( 0 ) = 0$, and denoting $a = d \varphi ( 0 )$, such that all eigenvalues of $a$ have strictly negative real part. Let $b(x,y) = \int_0^{+\infty} \langle e^{ta}(x) \mid e^{ta}(y) \rangle\, dt$ and $q(x) = b(x,x)$. Let $x_0 \in \mathbf{R}^n$ and $f_{x_0}$ the solution of $y' = \varphi(y),\ y(0) = x_0$. For any function $y$ defined on $\mathbf{R}_+$, define $\varepsilon(y)(t) = \varphi(y(t)) - a(y(t))$.
Prove the existence of two strictly positive real numbers $\alpha$ and $\beta$ such that, for all $t \in \mathbf { R } _ { + }$, we have: $$q \left( f _ { x _ { 0 } } ( t ) \right) \leqslant \alpha \Rightarrow - \left\| f _ { x _ { 0 } } ( t ) \right\| ^ { 2 } + 2 b \left( f _ { x _ { 0 } } ( t ) , \varepsilon \left( f _ { x _ { 0 } } \right) ( t ) \right) \leqslant - \beta q \left( f _ { x _ { 0 } } ( t ) \right)$$

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
We assume that $(\mu_{n})_{n \geqslant 1}$ is a sequence of strictly positive real numbers such that $\sum_{n \geqslant 1} \mu_{n}$ converges. We fix $\psi_{0} \in \mathcal{C}_{c}(\mathbb{R})$ and we define by recursion the sequence $(\psi_{n})_{n \geqslant 0}$ by
$$\forall n \geqslant 0,\quad \psi_{n+1} = T_{\mu_{n+1}} \psi_{n}$$
Show that for all $k \geqslant 1$ we have
$$\left\|\psi_{k}^{(k)}\right\|_{\infty} \leqslant \left\|\psi_{0}\right\|_{\infty} \frac{1}{\mu_{1} \cdots \mu_{k}}$$

In this part we consider a map $\varphi$ from $\mathbf { R } ^ { n }$ to $\mathbf { R } ^ { n }$ of class $\mathcal { C } ^ { 1 }$ such that $\varphi ( 0 ) = 0$, and denoting $a = d \varphi ( 0 )$, such that all eigenvalues of $a$ have strictly negative real part. Let $b(x,y) = \int_0^{+\infty} \langle e^{ta}(x) \mid e^{ta}(y) \rangle\, dt$ and $q(x) = b(x,x)$. Let $x_0 \in \mathbf{R}^n$ and $f_{x_0}$ the solution of $y' = \varphi(y),\ y(0) = x_0$. We fix strictly positive real numbers $\alpha$ and $\beta$ such that for all $t \in \mathbf{R}_+$: $$q(f_{x_0}(t)) \leqslant \alpha \Rightarrow -\|f_{x_0}(t)\|^2 + 2b(f_{x_0}(t), \varepsilon(f_{x_0})(t)) \leqslant -\beta q(f_{x_0}(t))$$
Show then that: $$q \left( x _ { 0 } \right) < \alpha \quad \Rightarrow \quad \forall t \geqslant 0 , q \left( f _ { x _ { 0 } } \right) ( t ) \leqslant e ^ { - \beta t } q \left( x _ { 0 } \right) .$$

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
We assume that $(\mu_{n})_{n \geqslant 1}$ is a sequence of strictly positive real numbers such that $\sum_{n \geqslant 1} \mu_{n}$ converges. We fix $\psi_{0} \in \mathcal{C}_{c}(\mathbb{R})$ and we define by recursion the sequence $(\psi_{n})_{n \geqslant 0}$ by
$$\forall n \geqslant 0,\quad \psi_{n+1} = T_{\mu_{n+1}} \psi_{n}$$
Let $f = \lim_{n \rightarrow \infty} \psi_{n}$. Show that $f$ has compact support and that if $\psi_{0}$ is positive and not identically zero, then so is $f$.