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In this paper we study properties of endomorphisms of $\p^k$ using asymmetric product construction $(\p^1)^k/\mathfrak{S}_k \cong \p^k$. Symmetricproducts have been used to produce examples of endomorphisms of $\p^k$ withcertain characteristics, $k\geq2$. In the present note, we discuss the use ofthese maps to enlighten stability phenomena in parameter spaces. In particular,we study $k$-deep post-critically finite maps and characterize families ofLatt\`es maps.

Symmetrization of Rational Maps: Arithmetic Properties and Families of Latt\`es Maps of $\mathbb P^k$