Abstract: We give a review of work on classifying rational proper maps between balls. The rational proper maps of degree 3 between $\mathbb{B}_2$ and $\mathbb{B}_4$ are still unclassified, which we study in this work. We find necessary equations to compute the Lebl normal forms of the known maps. We introduce two one-parameter families of such maps that are spherically inequivalent to one another and to any polynomial maps between $\mathbb{B}_2$ and $\mathbb{B}_4$.
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