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Register a new userGIT Stability of Henon Maps
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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 $operatorname{Rat}_2^2$.
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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.
We show that the dynamics of sufficiently dissipative semi-Siegel complex Henon maps with golden-mean rotation number is not $J$-stable in a very strong sense. By the work of Dujardin and Lyubich, this implies that the Newhouse phenomenon occurs for a dense $G_delta$ set of parameters in this family. Another consequence is that the Julia sets of such maps are disconnected for a dense set of parameters.
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.
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