Special Curves and Postcritically Finite Polynomials

Abstract

We 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 P 1 containing a Zariski-dense subset of postcritically finite maps.2010 Mathematics Subject Classification: 37F45 (primary); 11G50, 30C10 (secondary) OverviewIn this paper we address the question 'Which algebraic subvarieties of the moduli space M d 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 : P 1 → P 1 ; that is, every critical point of f has finite forward orbit under iteration. Postcritically finite (PCF) maps play an important role in complex dynamics, and in recent years there has been an explosion of work around them. It has been known since the foundational work of Thurston based on Teichmüller theory that PCF maps come in two flavors, the flexible Lattès maps (associated with elliptic curves and arising in one-dimensional families) and the rest (which are rigid). The rigid PCF maps form a countable Zariski-dense subset of the moduli space M d . From an arithmetic point of view, PCF maps are in several ways similar to elliptic curves with complex multiplication, and one can therefore view the above question as a dynamical analog of the André-Oort conjecture in arithmetic geometry. We formulate a conjectural answer to the above question: the special subvarieties V having a dense set of special points should M. Baker and L. DeMarco 2 be those for which the number of dynamically independent critical points does not exceed the dimension of V.As evidence for our general conjecture, we study rational curves within the parameter space of critically marked, monic, centered, degree-d polynomials. 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 in fact 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. We also study the family of curves Per 1 (λ) inside the space of cubic polynomials. First introduced by Milnor, Per 1 (λ) is defined as the set of maps with a fixed point of multiplier λ. The curve Per 1 (0), defined by the condition that one critical point is fixed, is special, and we prove that Per 1 (λ) is not special for all λ = 0.The proofs of these results rely on several ingredients, including: (1) an arithmetic equidistr...

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