Related papers: The Critical Locus for Complex H\'{e}non Maps

For maps of one complex variable, $f$, given as the sum of a degree $n$ power map and a degree $d$ polynomial, we provide necessary and sufficient conditions that the geometric limit as $n$ approaches infinity of the set of points that…

It is shown that critical phenomena associated with Siegel disk, intrinsic to 1D complex analytical maps, survives in 2D complex invertible dissipative H\'{e}non map. Special numerical method of estimation of the Siegel disk scaling center…

We study the post-critical set of a class of holomorphic systems with an irrationally indifferent fixed point. We prove a trichotomy for the topology of the post-critical set based on the arithmetic of the rotation number at the fixed…

The purpose of this note is two fold. First, we study the relation between a pair of H\'{e}non maps that share the same forward and backward non-escaping sets. Second, it is shown that there exists a continuum of $Short-\mathbb{C}^2$'s that…

Transcendental H\'enon maps are the natural extensions of the well investigated complex polynomial H\'enon maps to the much larger class of holomorphic automorphisms. We prove here that transcendental H\'enon maps always have non-trivial…

This paper constructs a combinatorial model for all postcritically finite rational maps arising as the Newton's method of a complex polynomial. This model is used in [LMS] to give a combinatorial classification of postcritically finite…

In this paper we build a geometric model for the renormalisation of irrationally indifferent fixed points of holomorphic maps with two critical points. The model incorporates arithmetic properties of the rotation number at the fixed point,…

For a hyperbolic polynomial automorphism of C^2 with a disconnected Julia set, and under a mild dissipativity condition, we give a topological description of the components of the Julia set. Namely, there are finitely many "quasi-solenoids"…

In complex dynamics, a fundamental result of Fatou and Julia asserts that every attracting cycle of a rational map attracts a critical point. The analogous statement fails in non-Archimedean dynamics. For a non-Archimedean rational map,…

We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials as dynamical systems. This lays the foundation for classification results of more general classes of Newton maps. A fundamental…

A cubic polynomial $P$ with a non-repelling fixed point $b$ is said to be immediately renormalizable if there exists a (connected) QL invariant filled Julia set $K^*$ such that $b\in K^*$. In that case, exactly one critical point of $P$…

We give an estimate of the discontinuity of the large Julia set for a perturbation of a class of maps tangent to the identity, by means of a two-dimensional Lavaurs Theorem. We adapt to our situation a strategy due to Bedford, Smillie and…

We give an alternative proof of the Benedicks-Carleson theorem on the existence of strange attractors in H\'enon-like families in the plane. To bypass a huge inductive argument, we introduce an induction-free explicit definition of…

In this paper, we construct geometrically finite rational maps with buried critical points on the boundaries of some hyperbolic components by using the pinching and plumbing deformations.

In this paper, we consider critical points of the horizontal energy $E_{\HH}(f)$ for a smooth map $f$ between two Riemannian foliations. These critical points are referred to as horizontally harmonic maps. In particular, if the maps are…

We use the homological perturbation lemma to produce explicit formulas computing the class in the twisted de Rham complex represented by an arbitrary polynomial. This is a non-asymptotic version of the method of Feynman diagrams. In…

We study numerically the $\alpha$- and $\omega$-limits of the Newton maps of two of the most elementary families of polynomial transformations on the plane: those with a linear component and those with both components of degree two. Our…

Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called \emph{dendritic}. By results of Kiwi, any dendritic polynomial is semi-conjugate to a topological polynomial whose topological Julia set…

Critical points of approximations of the Dirichlet energy \`{a} la Sacks-Uhlenbeck are known to converge to harmonic maps in a suitable sense. However, we show that not every harmonic map can be approximated by critical points of such…

The critical loci of a map $f:X\to Y$ between smooth schemes over a field $k$ are the locally closed subschemes $\Sigma^i(f)\subseteq X$ where the differential of $f$ has constant rank. We prove that if $f : X\to \mathbb A^r$ is the general…