Abstract
We develop techniques for using compactifications of Hurwitz spaces to study families of rational maps P 1 → P 1 defined by critical orbit relations. We apply these techniques in two settings: We show that the parameter space Per d , 4 of degree-d bicritical maps with a marked 4-periodic critical point is a d 2-punctured Riemann surface of genus ( d − 1 ) ( d − 2 ) 2 . We also recover a result of Canci and Vishkautsan, showing that the parameter space Per 2 , 5 of degree-2 rational maps with a marked 5-periodic critical point is a 10-punctured elliptic curve, and identifying its isomorphism class over Q . We carry out an experimental study of the interaction between dynamically defined points of Per 2 , 5 (such as PCF points or punctures) and the group structure of the underlying elliptic curve.
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