Equidistribution towards the bifurcation current I : Multipliers and degree d polynomials

In the moduli space of degree d polynomials, the set Per\_n(w) of classes [f] for which f admits a cycle of exact period n and multiplier w is known to be an algebraic hypersurface. We prove that, given a complex number w, these hypersurfaces equidistribute towards the bifurcation current as n tends to infinity.Comment: Modified third section. Some results of Section 3 have been made more precise. Minor errors correcte

Strong bifurcation loci of full Hausdorff dimension

In the moduli space $\mathcal{M}_d$ of degree $d$ rational maps, the bifurcation locus is the support of a closed $(1,1)$ positive current T_\bif which is called the bifurcation current. This current gives rise to a measure \mu_\bif:=(T_\bif)^{2d-2} whose support is the seat of strong bifurcations. Our main result says that \supp(\mu_\bif) has maximal Hausdorff dimension $2(2d-2)$. As a consequence, the set of degree $d$ rational maps having $2d-2$ distinct neutral cycles is dense in a set of full Hausdorff dimension.Comment: 37 page

Continuity of the Green function in meromorphic families of polynomials

We prove that along any marked point the Green function of a meromorphic family of polynomials parameterized by the punctured unit disk explodes exponentially fast near the origin with a continuous error term.Comment: Modified references. Added a corollary about the adelic metric associated with an algebraic family endowed with a marked poin

Symmetrization of Rational Maps: Arithmetic Properties and Families of Latt\`es Maps of Pk\mathbb P^k

In this paper we study properties of endomorphisms of $\mathbb P^k$ using a symmetric product construction $(\mathbb P^1)^k/\mathfrak{S}_k \cong \mathbb P^k$. Symmetric products have been used to produce examples of endomorphisms of $\mathbb P^k$ with certain characteristics, $k\geq2$. In the present note, we discuss the use of these maps to enlighten arithmetic phenomena and stability phenomena in parameter spaces. In particular, we study notions of uniform boundedness of rational preperiodic points via good reduction information, $k$-deep postcritically finite maps, and characterize families of Latt\`es maps.Comment: Added more background and references; repaired a small gap in Lemma 3.1; reordered some statements in Propositions 1.1 and 1.2; 26 page