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Rational maps with a preperiodic critical point

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arxiv 1806.11221 v2 pith:IBJSJCL4 submitted 2018-06-28 math.DS

classification math.DS
keywords criticalpointalgebraicclassesconjugacymapspolynomialspreperiodic
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abstract

We show that the set of conjugacy classes of cubic polynomials with a prefixed critical point, of preperiod $k\geq 1$, is an irreducible algebraic curve. We also establish an analogous result for quadratic rational maps. We then study a closely related question concerning the irreducibility (over $\mathbb Q$) of the set of conjugacy classes of unicritical polynomials, of degree $D\geq 2$, with a preperiodic critical point. Our proofs are purely algebraic.

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  1. A note on Misiurewicz polynomials

    math.NT 2019-08 conditional novelty 7.0 of 10

    For prime d, the Misiurewicz polynomial G_{d,m,n} has no more rational irreducible factors than its reduction modulo d, which yields new irreducibility families and the first bounds for periods above three.

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