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

Complex perspective for the projective heat map acting on pentagons

We place Schwartz's work on the real dynamics of the projective heat map H into the complex perspective by computing its rst dynamical degree and gleaning some corollaries about the dynamics of H

Geometric Limits of Julia Sets of Maps z^n + exp(2πiθ) as n → ∞

We show that the geometric limit as n → ∞ of the Julia sets J(Pn,c) for the maps Pn,c(z) = zn + c does not exist for almost every c on the unit circle. Furthermore, we show that there is always a subsequence along which the limit does exist and equals the unit circle

Superstable manifolds of invariant circles and codimension-one Böttcher functions

Let f : X ⇢ X be a dominant meromorphic self-map, where X is a compact, connected complex manifold of dimension n>1. Suppose that there is an embedded copy of P1 that is invariant under f, with f holomorphic and transversally superattracting with degree a in some neighborhood. Suppose that f restricted to this line is given by z↦zb, with resulting invariant circle S. We prove that if a≥b, then the local stable manifold Wsloc(S) is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show that the condition a≥b cannot be relaxed without adding additional hypotheses by presenting two examples with a<b for which Wsloc(S) is not real analytic in the neighborhood of any point

Results and Examples Regarding Bifurcation with a Two-Dimensional Kernal

Many problems in pure and applied mathematics entail studying the structure of solutions to F(x; y) = 0, where F is a nonlinear operator between Banach spaces and y is a real parameter. A parameter value where the structure of solutions of F changes is called a bifurcation point. The particular method of analysis for bifurcation depends on the dimension of the kernel of DxF(0,λ), the linearization of F. The purpose of our study was to examine some consequences of a recent theorem on bifurcations with 2-dimensional kernels. This resent theorem was compared to previous methods. Also, some specific classes of equations were identified in which the theorem always holds, and an algebraic example was found that illustrates bifurcations with a 2-dimensional kernel

Rational Map of CP^2 with No Invariant Foliation

Conference Poster presented at: Midwest Dynamical Systems Conference, Champaign/Urbana, IL November 1-3, 2013

Rational Maps of CP^2 with No Invariant Foliation

We present simple examples of rational maps of the complex projective plane with equal first and second dynamical degrees and no invariant foliation