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...

Abstract. We study critical orbits and bifurcations within the moduli space M 2 of quadratic rational maps, f : P 1 → P 1 . We focus on the family of curves, Per 1 (λ) ⊂ M 2 for λ ∈ C, defined by the condition that each f ∈ Per 1 (λ) has a fixed point of multiplier λ. We prove that the curve Per 1 (λ) contains infinitely many postcritically-finite maps if and only if λ = 0, addressing a special case of [BD2, Conjecture 1.4]. We also show that the two critical points of f define distinct bifurcation measures along Per 1 (λ).

We show that the weak limit of the maximal measures for any degenerating sequence of rational maps on the Riemann sphereĈ must be a countable sum of atoms. For a one-parameter family f t of rational maps, we refine this result by showing that the measures of maximal entropy have a unique limit onĈ as the family degenerates. The family f t may be viewed as a single rational function on the Berkovich projective line P 1 L over the completion of the field of formal Puiseux series in t, and the limiting measure onĈ is the 'residual measure' associated with the equilibrium measure on P 1 L . 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 on P 1 L .

We present a dynamical proof of the well-known fact that the Néron–Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field $k=\mathbb{C}(X)$, where $X$ is a curve. More generally, we investigate the mechanism by which the local canonical height for a map $f:\mathbb{P}^{1}\rightarrow \mathbb{P}^{1}$ defined over a function field $k$ can take irrational values (at points in a local completion of $k$), providing examples in all degrees $\deg f\geq 2$. Building on Kiwi’s classification of non-archimedean Julia sets for quadratic maps [Puiseux series dynamics of quadratic rational maps. Israel J. Math.201 (2014), 631–700], we give a complete answer in degree 2 characterizing the existence of points with irrational local canonical heights. As an application we prove that if the heights $\widehat{h}_{f}(a),\widehat{h}_{g}(b)$ are rational and positive, for maps $f$ and $g$ of multiplicatively independent degrees and points $a,b\in \mathbb{P}^{1}(\bar{k})$, then the orbits $\{f^{n}(a)\}_{n\geq 0}$ and $\{g^{m}(b)\}_{m\geq 0}$ intersect in at most finitely many points, complementing the results of Ghioca et al [Intersections of polynomials orbits, and a dynamical Mordell–Lang conjecture. Invent. Math.171 (2) (2008), 463–483].

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ f_c(z) = z^2+c $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ c \in {\mathbb C} $\end{document}</tex-math></inline-formula>. We show there exists a uniform upper bound on the number of points in <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb P}^1( {\mathbb C}) $\end{document}</tex-math></inline-formula> that can be preperiodic for both <inline-formula><tex-math id="M4">\begin{document}$ f_{c_1} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ f_{c_2} $\end{document}</tex-math></inline-formula>, for any pair <inline-formula><tex-math id="M6">\begin{document}$ c_1\not = c_2 $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M7">\begin{document}$ {\mathbb C} $\end{document}</tex-math></inline-formula>. The proof combines arithmetic ingredients with complex-analytic: we estimate an adelic energy pairing when the parameters lie in <inline-formula><tex-math id="M8">\begin{document}$ \overline{\mathbb{Q}} $\end{document}</tex-math></inline-formula>, building on the quantitative arithmetic equidistribution theorem of Favre and Rivera-Letelier, and we use distortion theorems in complex analysis to control the size of the intersection of distinct Julia sets. The proofs are effective, and we provide explicit constants for each of the results.</p>