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Construction of Rational Maps on the Projective Line with Given Dynamical Structure

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Technical Report

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U.S. Naval Academy Annapolis United States

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In this paper we prove that we can construct a unique quadratic rational map on the projective line if given three fixed points and a pairof period two points. There are restrictions on the given points related to maintaining distinct existence of the fixed and periodic points.We construct the quadratic rational map by focusing on the case of fixed points at 0, 1, infinity. In this space we use a Grobner basis to solve a system of equations formed by the coefficients of fixed point polynomials. The solution to this system is the set of coefficients of the quadratic rational map. Using a Mobius transformation, we can send any three distinct, desired fixed points to 0, 1, infinity, construct the map, and use an inverse Mobius transformation to bring the map to the original fixed points. As an application we discuss constructing certain elliptic curves via Lattes maps.

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