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Register a new userDynamical modular curves for quadratic polynomial maps
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Authors
John R. Doyle
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Motivated by the dynamical uniform boundedness conjecture of Morton and Silverman, specifically in the case of quadratic polynomials, we give a formal construction of a certain class of dynamical analogues of classical modular curves. The preperiodic points for a quadratic polynomial map may be endowed with the structure of a directed graph satisfying certain strict conditions; we call such a graph admissible. Given an admissible graph $G$, we construct a curve $X_1(G)$ whose points parametrize quadratic polynomial maps -- which, up to equivalence, form a one-parameter family -- together with a collection of marked preperiodic points that form a graph isomorphic to $G$. Building on work of Bousch and Morton, we show that these curves are irreducible in characteristic zero, and we give an application of irreducibility in the setting of number fields. We end with a discussion of the Galois theory associated to the preperiodic points of quadratic polynomials, including a certain Galois representation that arises naturally in this setting.
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Given a number field $K$ and a polynomial $f(z) in K[z]$ of degree at least 2, one can construct a finite directed graph $G(f,K)$ whose vertices are the $K$-rational preperiodic points for $f$, with an edge $alpha to beta$ if and only if $f(alpha) = beta$. Restricting to quadratic polynomials, the dynamical uniform boundedness conjecture of Morton and Silverman suggests that for a given number field $K$, there should only be finitely many isomorphism classes of directed graphs that arise in this way. Poonen has given a conjecturally complete classification of all such directed graphs over $mathbb{Q}$, while recent work of the author, Faber, and Krumm has provided a detailed study of this question for all quadratic extensions of $mathbb{Q}$. In this article, we give a conjecturally complete classification like Poonens, but over the cyclotomic quadratic fields $mathbb{Q}(sqrt{-1})$ and $mathbb{Q}(sqrt{-3})$. The main tools we use are dynamical modular curves and results concerning quadratic points on curves.
In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of $(2:2)$ holomorphic correspondences $mathcal{F}_a$: $$left(frac{aw-1}{w-1}right)^2+left(frac{aw-1}{w-1}right)left(frac{az+1}{z+1}right) +left(frac{az+1}{z+1}right)^2=3$$ and proved that for every value of $a in [4,7] subset mathbb{R}$ the correspondence $mathcal{F}_a$ is a mating between a quadratic polynomial $Q_c(z)=z^2+c,,,c in mathbb{R}$ and the modular group $Gamma=PSL(2,mathbb{Z})$. They conjectured that this is the case for every member of the family $mathcal{F}_a$ which has $a$ in the connectedness locus. We prove here that every member of the family $mathcal{F}_a$ which has $a$ in the connectedness locus is a mating between the modular group and an element of the parabolic quadratic family $Per_1(1)$.
We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an unlikely intersection statement for such pairs thereby exhibiting strong rigidity features for these pairs. We infer from this result the dynamical Andre-Oort conjecture for curves in the moduli space of polynomials, by describing one-dimensional families in this parameter space containing infinitely many post-critically finite parameters.
In this paper we study the locus of generalized degree $d$ Henon maps in the parameter space $operatorname{Rat}_d^N$ of degree $d$ rational maps $mathbb{P}^Ntomathbb{P}^N$ modulo the conjugation action of $operatorname{SL}_{N+1}$. We show that Henon maps are in the GIT unstable locus if $Nge3$ or $dge3$, and that they are semistable, but not stable, in the remaining case of $N=d=2$. We also give a general classification of all unstable maps in $operatorname{Rat}_2^2$.
We prove that there exists a homeomorphism $chi$ between the connectedness locus $mathcal{M}_{Gamma}$ for the family $mathcal{F}_a$ of $(2:2)$ holomorphic correspondences introduced by Bullett and Penrose, and the parabolic Mandelbrot set $mathcal{M}_1$. The homeomorphism $chi$ is dynamical ($mathcal{F}_a$ is a mating between $PSL(2,mathbb{Z})$ and $P_{chi(a)}$), it is conformal on the interior of $mathcal{M}_{Gamma}$, and it extends to a homeomorphism between suitably defined neighbourhoods in the respective one parameter moduli spaces. Following the recent proof by Petersen and Roesch that $mathcal{M}_1$ is homeomorphic to the classical Mandelbrot set $mathcal{M}$, we deduce that $mathcal{M}_{Gamma}$ is homeomorphic to $mathcal{M}$.
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