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\begin{document}


\begin{center}
\textbf{\huge{Teor\'ia Geom\'etrica de la Medida}}
\end{center}

\begin{center}
 \bf{\LARGE{Ejercicios}}
\end{center}

\begin{description}

\item[Ejercicio 23:] (Clase 12, 29/05)

Probar que $\Sigma_m$ es totalmente disconexo con la m\'etrica
definida por:
$$\rho(\alpha,\beta)=\sum_{i=1}^{+\infty}\frac{|\alpha_i-\beta_i|}{m^i}.$$

\item[Ejercicio 24:] (Clase 12, 29/05)

Sea $\phi=\{\varphi_1,\ldots,\p_m\}$, con
$\varphi_i:\R^d\longrightarrow\R^d$ contractivas e inyectivas y sea $K$ el
conjunto autosimilar asociado a $\phi$. \\
Si $\varphi_i(K)\cap\varphi_j(K)=\emptyset$ para $i\neq j$, probar
que $K$ es homeomorfo a $\Sigma_m$ y que es totalmente disconexo.

\item[Ejercicio 25:] (Clase 13, 31/05)

Sea $(X,d)$ un espacio m\'etrico y sean  $A,B$ cerrados disjuntos de
$X$. Probar que existe una funci\'on $f:X\longrightarrow \R$
continua, tal que: $$f\mid_A\equiv0,\quad f\mid_B\equiv1\quad\text{
y }\quad0\leq f(x)\leq1\quad\forall\, x\in X.$$ Demostrar adem\'as,
que si $\delta:=d(A,B)>0$, entonces $f$ es Lipschitz con constante
$\frac{3}{\delta}$.

\end{description}

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