Preperiodic points and unlikely intersections

By Matthew Baker, Laura Demarco. 2009

123 citations recorded by the source corpus.

In this article, we combine complex-analytic and arithmetic tools to study the preperiodic points of one-dimensional complex dynamical systems. We show that for any fixed a, b ∈ C, and any integer d ≥ 2, the set of c ∈ C for which both a and b are preperiodic for z + c is infinite if and only if a = b. This provides an affirmative answer to a question of Zannier, which itself arose from questions of Masser concerning simultaneous torsion sections on families of elliptic curves. Using similar techniques, we prove that if rational functions φ,ψ ∈ C(z) have infinitely many preperiodic points in common, then all of their preperiodic points coincide (and in particular the maps must have the same Julia set). This generalizes a theorem of Mimar, who established the same result assuming that φ and ψ are defined over Q. The main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields, with non-archimedean Berkovich spaces playing an essential role.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/10ff44514f519e23e6009d03c1fc3c20e081e101

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Citations in the corpus

Bifurcations, intersections, and heights (2014, result): “We expect that (1) =⇒ (2) should be a consequence of “arithmetic equidistribution” as in the proofs of [Baker and DeMarco 2011; 2013; Ghioca et al. 2013; 2015; DeMarco et al. 2016] when V is a curve.”
Dynamics on Berkovich Spaces in Low Dimensions (2012, methodology): “original dynamical system is dened over the complex numbers. We shall see some instances of this phenomenon in these notes; other examples are provided by the work of Kiwi [Kiw06], Baker and DeMarco [BdM09], and Ghioca, Tucker and Zieve [GTZ08]. Over the complex numbers, many of the most powerful to”
Current trends and open problems in arithmetic dynamics (2018, result): “Building on a number of earlier partial results [8,9,70,89,96], Favre and Gauthier have recently announced a proof of Conjecture 11.”