The present thesis has two main parts. In the first one, we study bounds for the number of rational preperiodic points of an endomorphism of $\PP^1$. Let $K$ be a number field and $\phi$ be an endomorphism of $\PP^1$ over $K$ of degree $d\geq 2$. Let $S$ be the set of places of bad reduction for $\phi$ (including the archimedean places). Let $\Per(\phi,K)$, $\PrePer(\phi, K)$, and $\Tail(\phi,K)$ be the set of $K$-rational periodic, preperiodic, and purely preperiodic points of $\phi$, respectively.If we assume that $|\Per(\phi,K)| \geq 4$ (resp.\ $|\Tail(\phi,K)| \geq 3$), we prove bounds for $|\Tail(\phi,K)|$ (resp.\ $|\Per(\phi,K)|$) that depend only on the number of places of bad reduction $|S|$ (and not on the degree $d$). We show that the hypotheses of this result are sharp, giving counterexamples to any possible result of this form when $|\Per(\phi,K)| < 4$ (resp.\ $|\Tail(\phi,K)| < 3$). The key tool involved in these results is a bound for the number of solutions of $S$-unit equations.Using bounds for the number of solutions of the celebrated Thue-Mahler equation, we obtain bounds for $|\Per(\phi,K)|$ and $|\Tail(\phi,K)|$ in terms of the number of places of bad reduction $|S|$ and the degree $d$ of the rational function $\phi$. Bounds obtained in this way are a significant improvement to previous result given by J. Canci and L. Paladino.In the second part of the thesis, we study the set of $K$-rational purely preperiodic hypersurfaces of $\PP^n$ of a given degree for an endomorphism of $\PP^n$. Let $\phi$ be an endomorphism of $\PP^n$ over $K$, $S$ be the set of places of bad reduction for $\phi$ and $\HTail(\phi,K,e)$ be the set of $K$-rational purely preperiodic hypersurfaces of $\PP^n$ of degree $e$. We give a strong arithmetic relation between $K$-rational purely preperiodic hypersurfaces and $K$-rational periodic points. If we consider $N=\binom{e+n}{e}-1$ and assume that $\phi$ has at least $2N+1$ $K$-rational periodic points such that no $N+1$ of them lie in a hypersurface of degree $e$ then we give an effective bound on a large subset of $\HTail(\phi,K,e)$ depending on $e$ and the number of places of bad reduction $|S|$. Finally, we prove that the set $\HTail(\phi,K,e)$ is finite if we assume that $\phi$ is an endomorphism of $\PP^2$.
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