Abstract
AbstractWe study the postcritically finite maps within the moduli space of complex polynomial dynamical systems. We characterize rational curves in the moduli space containing an infinite number of postcritically finite maps, in terms of critical orbit relations, in two settings: (1) rational curves that are polynomially parameterized; and (2) cubic polynomials defined by a given fixed point multiplier. We offer a conjecture on the general form of algebraic subvarieties in the moduli space of rational maps on ${ \mathbb{P} }^{1} $ containing a Zariski-dense subset of postcritically finite maps.
Highlights
For each integer d ≥ 2, let MPcdm denote the moduli space of critically marked complex polynomials of degree d
Overview In this paper we address the question ‘Which algebraic subvarieties of the moduli space Md of degree-d rational maps contain a Zariski-dense set of special points?’ Here, a special point is the conjugacy class of a postcritically finite map f : P1 → P1; that is, every critical point of f has finite forward orbit under iteration
Our main result provides an explicit description of the polynomially parameterized rational curves that are special: they are those for which there is exactly one active critical orbit, up to polynomial symmetries. (We prove a more general result about marked but not necessarily critical points that are simultaneously preperiodic.) We provide examples showing that the ‘up to symmetries’ condition is necessary, and we illustrate how one can check that a given curve is special
Summary
For each integer d ≥ 2, let MPcdm denote the moduli space of critically marked complex polynomials of degree d. By a classical complex dynamics argument, the active critical point must have finite forward orbit for a dense set of parameters in the bifurcation locus, so there are infinitely many PCF polynomials f ∈ Per1(0). We expect that an algebraic subvariety V in MPcdm contains a Zariski-dense subset of PCF maps if and only if V is cut out by critical orbit relations. PCF maps, where both critical points have finite forward orbit, by condition (3) of Theorem 1.2. Let {ft: t ∈ V} be an N-dimensional algebraic family of critically marked rational maps of degree d ≥ 2. One implication of Conjecture 1.10 (dynamical dependence of any active (N + 1)-tuple of critical points implies Zariski density of PCF maps) follows from an argument mimicking the proof of Proposition 2.6 and the following observation. By Thurston’s rigidity theorem, this is the only positive-dimensional family with no active critical points; see [Mc1, Theorem 2.2], [DH]
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