On some exceptional rational maps

“…This question was completely answered by Baker [1] in the case that M = 0. (See also [9,Thm. 1] for the corresponding statement for rational functions.)…”

Section: Introductionmentioning

confidence: 99%

Preperiodic portraits for unicritical polynomials

Doyle

2016

Proc. Amer. Math. Soc.

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Let K be an algebraically closed field of characteristic zero, and for c ∈ K and an integer d ≥ 2,We consider the following question: If we fix x ∈ K and integers M ≥ 0, N ≥ 1, and d ≥ 2, does there exist c ∈ K such that, under iteration by f d,c , the point x enters into an N -cycle after precisely M steps? We conclude that the answer is generally affirmative, and we explicitly give all counterexamples. When d = 2, this answers a question posed by Ghioca, Nguyen, and Tucker.(X 2 +C) 2 +C−X X 2 +C−X = X 2 + X + C + 1 = 0, illustrated in Figure 1. The bifurcation at c = −3/4 may be seen by letting c tend to −3/4 and observing that the two points on Y 2 lying over c (corresponding to the two points of period 2 for z 2 + c) approach a single point on Y 1 (corresponding to a fixed point for z 2 − 3/4).Baker showed that the polynomial ϕ(z) = z 2 − 3/4 is, in some sense, the only polynomial of degree at least 2 that fails to admit points of a given period. To make this more precise, we first recall the following

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“…It was Faber and Granville [7] who pointed out that the condition n / ∈ X(ϕ) would be necessary. By Lemma 4.24, n ∈ X(ϕ) if and only if either ϕ has no point of minimum period n (see Kisaka's classification [14] or [7,Appendix B] for a complete list of all rational maps which have no point of minimum period n), or n = 2 and ϕ(z) is linearly conjugate to z −2 . Note that Ingram-Silverman made their conjectures over number fields while Faber-Granville even restricted further to the field of rational numbers.…”

Section: Introductionmentioning

confidence: 99%

Portraits of preperiodic points for rational maps

Ghioca

Nguyen

Tucker

2015

Math. Proc. Camb. Phil. Soc.

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Let $K$ be a function field over an algebraically closed field $k$ of characteristic $0$, let $\varphi\in K(z)$ be a rational function of degree at least equal to $2$ for which there is no point at which $\varphi$ is totally ramified, and let $\alpha\in K$. We show that for all but finitely many pairs $(m,n)\in \mathbb{Z}_{\ge 0}\times \mathbb{N}$ there exists a place $\mathfrak{p}$ of $K$ such that the point $\alpha$ has preperiod $m$ and minimum period $n$ under the action of $\varphi$. This answers a conjecture made by Ingram-Silverman and Faber-Granville. We prove a similar result, under suitable modification, also when $\varphi$ has points where it is totally ramified. We give several applications of our result, such as showing that for any tuple $(c_1,\dots , c_{d-1})\in k^{n-1}$ and for almost all pairs $(m_i,n_i)\in \mathbb{Z}_{\ge 0}\times \mathbb{N}$ for $i=1,\dots, d-1$, there exists a polynomial $f\in k[z]$ of degree $d$ in normal form such that for each $i=1,\dots, d-1$, the point $c_i$ has preperiod $m_i$ and minimum period $n_i$ under the action of $f$

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On some exceptional rational maps

Cited by 6 publications

References 1 publication

Quadratic rational maps lacking period 2 orbits

Quadratic rational maps lacking period 2 orbits

Preperiodic portraits for unicritical polynomials

Portraits of preperiodic points for rational maps

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