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Laura DeMarco

Publications and source records attributed to Laura DeMarco.

32 records · Page 2Linked to original sources
arXiv2014

Torsion points and the Lattes family

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.

math.DS↗
arXiv2013

Special curves and postcritically-finite polynomials

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 $\mathrm{Per}_1(λ)$ in the space of cubic polynomials, introduced by Milnor (1992), we show that $\mathrm{Per}_1(λ)$ contains infinitely many PCF maps if and only if $λ=0$. 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 André-Oort Conjecture from arithmetic geometry.

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arXiv2013

Degenerations of Complex Dynamical Systems

We show that the weak limit of the maximal measures for any degenerating sequence of rational maps on the Riemann sphere must be a countable sum of atoms. For a 1-parameter family f_t of rational maps, we refine this result by showing that the measures of maximal entropy have a unique limit on the Riemann sphere as the family degenerates. The family f_t may be viewed as a single rational function on the Berkovich projective line over the completion of the field of formal Puiseux series in t, and the limiting measure on the Riemann sphere is the "residual measure" associated to the equilibrium measure on the Berkovich line. For the proof, we introduce a new technique for quantizing measures on the Berkovich projective line and demonstrate the uniqueness of solutions to a quantized version of the pullback formula for the equilibrium measure there.

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arXiv2012

The geometry of the critically-periodic curves in the space of cubic polynomials

We provide an algorithm for computing the Euler characteristic of the curves $S_p$ in the space of cubic polynomials, consisting of all polynomials with a periodic critical point of period $p$. The curves were introduced in [Milnor, Bonifant-Kiwi-Milnor], and the algorithm applies the main results of [DeMarco-Pilgrim]. The output is shown for periods $p \leq 26$.

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arXiv2011

Polynomial basins of infinity

We study the projection $π: M_d \to B_d$ 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 $π$ becomes continuous and a homeomorphism on the shift locus. Our main result is that all fibers of $π$ 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.

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arXiv2011

The classification of polynomial basins of infinity

We consider the problem of classifying the dynamics of complex polynomials $f: \mathbb{C} \to \mathbb{C}$ restricted to their basins of infinity. We synthesize existing combinatorial tools --- tableaux, trees, and laminations --- into a new invariant of basin dynamics we call the pictograph. For polynomials with all critical points escaping to infinity, we obtain a complete description of the set of topological conjugacy classes. We give an algorithm for constructing abstract pictographs, and we provide an inductive algorithm for counting topological conjugacy classes with a given pictograph.

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arXiv2010

Preperiodic points and unlikely intersections

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 complex numbers a and b, and any integer d at least 2, the set of complex numbers c for which both a and b are preperiodic for z^d+c is infinite if and only if a^d = b^d. 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 two complex rational functions f and g have infinitely many preperiodic points in common, then they must have the same Julia set. This generalizes a theorem of Mimar, who established the same result assuming that f and g are defined over an algebraic extension of the rationals. 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.

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arXiv2009

Critical heights on the moduli space of polynomials

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.

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arXiv2007

Axiom A polynomial skew products of C^2 and their postcritical sets

A polynomial skew product of C^2 is a map of the form f(z,w) = (p(z), q(z,w)), where p and q are polynomials, such that f is regular of degree d >= 2. For polynomial maps of C, hyperbolicity is equivalent to the condition that the closure of the postcritical set is disjoint from the Julia set; further, critical points either iterate to an attracting cycle or infinity. For polynomial skew products, Jonsson (Math. Ann., 1999) established that f is Axiom A if and only if the closure of the postcritical set is disjoint from the right analog of the Julia set. Here we present the analogous conclusion: critical orbits either escape to infinity or accumulate on an attracting set. In addition, we construct new examples of Axiom A maps demonstrating various postcritical behaviors.

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arXiv2006

Finiteness for degenerate polynomials

Let $\MP_d$ denote the space of polynomials $f: \C \to \C$ of degree $d\geq 2$, modulo conjugation by $\Aut(\C)$. Using properties of polynomial trees (as introduced in [DM, math.DS/0608759]), we show that if $f_n$ is a divergent sequence of polynomials in $\MP_d$, then any subsequential limit of the measures of maximal entropy $m(f_n)$ will have finite support. With similar techniques, we observe that the iteration maps $\{\MPbar_d \dashrightarrow \MPbar_{d^n}: n\geq 1\}$ between GIT-compactifications can be resolved simultaneously with only finitely many blow-ups of $\MPbar_d$.

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arXiv2006

Transfinite diameter and the resultant

We prove a formula for the Fekete-Leja transfinite diameter of the pullback of a set E in C^N by a regular polynomial map F, expressing it in terms of the resultant of the leading part of F and the transfinite diameter of E. We also establish the nonarchimedean analogue of this formula. A key step in the proof is a formula for the transfinite diameter of the filled Julia set of F.

math.CV↗
arXiv2004

Iteration at the boundary of the space of rational maps

Let $Rat_d$ denote the space of holomorphic self-maps of ${\bf P}^1$ of degree $d\geq 2$, and $μ_f$ the measure of maximal entropy for $f\in Rat_d$. The map of measures $f\mapstoμ_f$ 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.

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arXiv2004

The boundary of the moduli space of quadratic rational maps

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 Möbius 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.

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