Let $M_d$ be the moduli space of one-dimensional complex polynomial dynamical systems. The escape rates of the critical points determine a critical heights map $G: M_d \to \mathbb{R}^{d-1}$. For generic values of $G$, each connected component of a fiber of $G$ is the deformation space for twist deformations on the basin of infinity. We analyze the quotient space $\mathcal{T}_d^*$ obtained by collapsing each connected component of a fiber of $G$ to a point. The space $\mathcal{T}_d^*$ is a parameter-space analog of the polynomial tree $T(f)$ associated to a polynomial $f:\mathbb{C}\to\mathbb{C}$, studied by DeMarco and McMullen, and there is a natural projection from $\mathcal{T}_d^*$ to the space of trees $\mathcal{T}_d$. We show that the projectivization $\mathbb{P}\mathcal{T}_d^*$ is compact and contractible; further, the shift locus in $\mathbb{P}\mathcal{T}_d^*$ has a canonical locally finite simplicial structure. The top-dimensional simplices are in one-to-one corespondence with topological conjugacy classes of structurally stable polynomials in the shift locus.

Laura DeMarco Kevin Pilgrim

Let $Rat_d$ denote the space of holomorphic self-maps of ${\bf P}^1$ of degree $d\geq 2$, and $\mu_f$ the measure of maximal entropy for $f\in Rat_d$. The map of measures $f\mapsto\mu_f$ is known to be continuous on $Rat_d$, and it is shown here to extend continuously to the boundary of $Rat_d$ in $\bar{Rat}_d \simeq {\bf P}^{2d+1}$, except along a locus $I(d)$ of codimension $d+1$. The set $I(d)$ is also the indeterminacy locus of the iterate map $f\mapsto f^n$ for every $n\geq 2$. The limiting measures are given explicitly, away from $I(d)$. The degenerations of rational maps are also described in terms of metrics of non-negative curvature on the Riemann sphere: the limits are polyhedral.

Laura DeMarco

Let $M_2$ be the space of quadratic rational maps $f:{\bf P}^1\to{\bf P}^1$, modulo the action by conjugation of the group of M\"obius transformations. In this paper a compactification $X$ of $M_2$ is defined, as a modification of Milnor's $\bar{M}_2\iso{\bf CP}^2$, by choosing representatives of a conjugacy class $[f]\in M_2$ such that the measure of maximal entropy of $f$ has conformal barycenter at the origin in ${\bf R}^3$, and taking the closure in the space of probability measures. It is shown that $X$ is the smallest compactification of $M_2$ such that all iterate maps $[f]\mapsto [f^n]\in M_{2^n}$ extend continuously to $X \to \bar{M}_{2^n}$, where $\bar{M}_d$ is the natural compactification of $M_d$ coming from geometric invariant theory.

Laura DeMarco

The basin of infinity of a polynomial map $f : {\bf C} \arrow {\bf C}$ carries a natural foliation and a flat metric with singularities, making it into a metrized Riemann surface $X(f)$. As $f$ diverges in the moduli space of polynomials, the surface $X(f)$ collapses along its foliation to yield a metrized simplicial tree $(T,\eta)$, with limiting dynamics $F : T \arrow T$. In this paper we characterize the trees that arise as limits, and show they provide a natural boundary $\PT_d$ compactifying the moduli space of polynomials of degree $d$. We show that $(T,\eta,F)$ records the limiting behavior of multipliers at periodic points, and that any divergent meromorphic family of polynomials $\{f_t(z) : t \mem \Delta^* \}$ can be completed by a unique tree at its central fiber. Finally we show that in the cubic case, the boundary of moduli space $\PT_3$ is itself a tree. The metrized trees $(T,\eta,F)$ provide a counterpart, in the setting of iterated rational maps, to the ${\bf R}$-trees that arise as limits of hyperbolic manifolds.

Laura G. DeMarco Curtis T. McMullen

We study the projection $\pi: M_d \to B_d$ which sends an affine conjugacy class of polynomial $f: \mathbb{C}\to\mathbb{C}$ to the holomorphic conjugacy class of the restriction of $f$ to its basin of infinity. When $B_d$ is equipped with a dynamically natural Gromov-Hausdorff topology, the map $\pi$ becomes continuous and a homeomorphism on the shift locus. Our main result is that all fibers of $\pi$ are connected. Consequently, quasiconformal and topological basin-of-infinity conjugacy classes are also connected. The key ingredient in the proof is an analysis of model surfaces and model maps, branched covers between translation surfaces which model the local behavior of a polynomial.

Laura DeMarco Kevin Pilgrim

We study the postcritically-finite (PCF) maps in the moduli space of complex polynomials $\mathrm{MP}_d$. For a certain class of rational curves $C$ in $\mathrm{MP}_d$, we characterize the condition that $C$ contains infinitely many PCF maps. In particular, we show that if $C$ is parameterized by polynomials, then there are infinitely many PCF maps in $C$ if and only if there is exactly one active critical point along $C$, up to symmetries; we provide the critical orbit relation satisfied by any pair of active critical points. For the curves $\mathrm{Per}_1(\lambda)$ in the space of cubic polynomials, introduced by Milnor (1992), we show that $\mathrm{Per}_1(\lambda)$ contains infinitely many PCF maps if and only if $\lambda=0$. The proofs involve a combination of number-theoretic methods (specifically, arithmetic equidistribution) and complex-analytic techniques (specifically, univalent function theory). We provide a conjecture about Zariski density of PCF maps in subvarieties of the space of rational maps, in analogy with the Andr\'e-Oort Conjecture from arithmetic geometry.

Matthew Baker Laura DeMarco

We give a dynamical proof of a result of Masser and Zannier [MZ2, MZ3] about torsion points on the Legendre family of elliptic curves. Our methods also treat points of small height. A key ingredient is the arithmetic equidistribution theorem on $\mathbb{P}^1$ of Baker-Rumely, Chambert-Loir, and Favre-Rivera-Letelier. Torsion points on the elliptic curve coincide with preperiodic points for the degree-4 Lattes family of rational functions. Our main new results concern properties of the bifurcation measures for this Lattes family associated to marked points.

Laura DeMarco Xiaoguang Wang Hexi Ye

We measure the thermalization dynamics of a lattice Bose gas that is Stark localized by a parabolic potential. A non-equilibrium thermal density distribution is created by quickly removing an optical barrier. The resulting spatio-temporal dynamics are resolved using Mardia's $B$ statistic, which is a measure sensitive to the shape of the entire density distribution. We conclude that equilibrium is achieved for all lattice potential depths that we sample, including the strongly interacting and localized regime. However, thermalization is slow and non-exponential, requiring up to 500 tunneling times. We show that the Hubbard $U$ term is not responsible for thermalization via comparison to an exact diagonalization calculation, and we rule out equilibration driven by lattice-light heating by varying the laser wavelength. The thermalization timescale is comparable to the next-nearest-neighbor tunneling time, which suggests that a continuum, strongly interacting theory may be needed to understand equlibration in this system.

Laura Wadleigh Nicholas Kowalski Brian DeMarco

In [DKY], it was conjectured that there is a uniform bound $B$, depending only on the degree $d$, so that any pair of holomorphic maps $f, g :\mathbb{P}^1\to\mathbb{P}^1$ with degree $d$ will either share all of their preperiodic points or have at most $B$ in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, $\mathrm{Rat}_d \times \mathrm{Rat}_d$, for each degree $d\geq 2$. The proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier-Vigny, Yuan-Zhang, and Mavraki-Schmidt. In addition, we present alternate proofs of recent results of DeMarco-Krieger-Ye and of Poineau. In fact we prove a generalization of a conjecture of Bogomolov-Fu-Tschinkel in a mixed setting of dynamical systems and elliptic curves.

Laura DeMarco Niki Myrto Mavraki

Let a and b be algebraic numbers such that exactly one of a and b is an algebraic integer, and let f_t(z):=z^2+t be a family of polynomials parametrized by t. We prove that the set of all algebraic numbers t for which there exist positive integers m and n such that f_t^m(a)=f_t^n(b) has bounded Weil height. This is a special case of a more general result supporting a new bounded height conjecture in dynamics. Our results fit into the general setting of the principle of unlikely intersections in arithmetic dynamics.

Laura DeMarco Dragos Ghioca Holly Krieger Khoa D. Nguyen Thomas J. Tucker Hexi Ye