Abstract: We introduce and study rational m-fold sphere maps, that is, rational maps taking m spheres to m spheres. We show that a polynomial m-fold sphere map of degree m or less is an $\infty$-fold sphere map, that is, takes infinitely many spheres to spheres. Similarly, every rational m-fold sphere map of degree less than m is an $\infty$-fold sphere map. We then prove that $\infty$-fold sphere maps are, up to a unitary transformation, direct sums of finitely many homogeneous sphere maps.
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