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John Erik Fornaess

Publications and source records attributed to John Erik Fornaess.

At least 19 recordsLinked to original sources
arXiv2025

Orbits Inside Basins of Attraction of Skew Products

A basic problem in complex dynamics is to understand orbits of holomorphic maps. One problem is to understand the collection of points $S$ in an attracting basin whose forward orbits land exactly on the attracting fixed point. In the paper [13], the second author showed that for holomorphic polynomials in $\mathbb C$, there is a constant $C$ so that all Kobayashi discs of radius $C$ must intersect this set $S$. In the paper [15], the second author showed that there are holomorphic skew products in $\mathbb {C}^2$ where this result fails. The main result of this paper is to show that for a large class of polynomial skew products, this result nevertheless holds.

math.DS↗
arXiv2024

Backtracking New Q-Newton's method, Newton's flow, Voronoi's diagram and Stochastic root finding

A new variant of Newton's method - named Backtracking New Q-Newton's method (BNQN) - which has strong theoretical guarantee, is easy to implement, and has good experimental performance, was recently introduced by the third author. Experiments performed previously showed some remarkable properties of the basins of attractions for finding roots of polynomials and meromorphic functions, with BNQN. In general, they look more smooth than that of Newton's method. In this paper, we continue to experimentally explore in depth this remarkable phenomenon, and connect BNQN to Newton's flow and Voronoi's diagram. This link poses a couple of challenging puzzles to be explained. Experiments also indicate that BNQN is more robust against random perturbations than Newton's method and Random Relaxed Newton's method.

math.OC↗
arXiv2023

Backtracking New Q-Newton's method, Schröder's theorem, and Linear Conjugacy

A new variant of Newton's method - named Backtracking New Q-Newton's method (BNQN) - which has strong theoretical guarantee, is easy to implement, and has good experimental performance, was recently introduced by the third author. Experiments performed previously showed some remarkable properties of the basins of attractions for finding roots of polynomials and meromorphic functions using BNQN. In particular, it seems that for finding roots of polynomials of degree 2, the basins of attraction of the dynamics for BNQN are the same as that for Newton's method (the latter is the classical Schröder's result in Complex Dynamics). In this paper, we show that indeed the picture we obtain when finding roots of polynomials of degree 2 is the same as that in Schöder's result, with a remarkable difference: on the boundary line of the basins, the dynamics of Newton's method is chaotic, while the dynamics of BNQN is more smooth. On the way to proving the result, we show that BNQN (in any dimension) is invariant under conjugation by linear operators of the form $A=cR$, where $R$ is unitary and $c>0$ a constant. This again illustrates the similarity-difference relation between BNQN and Newton's method.

math.DS↗
arXiv2023

Orbits inside Fatou sets

In this paper, we investigate the precise behavior of orbits inside attracting basins of rational functions on $\mathbb P^1$ and entire functions $f$ in $\mathbb{C}$. Let $R(z)$ be a rational function on $\mathbb P^1$, $\mathcal {A}(p)$ be the basin of attraction of an attracting fixed point $p$ of $R$, and $Ω_i$ $ (i=1, 2, \cdots)$ be the connected components of $\mathcal{A}(p)$, and $Ω_1$ contains $p.$ Let $p_0\inΩ_1$ be close to $p.$ If at least one $Ω_i$ is not simply connected, then there exists a constant $C$ so that for any $z_0\in Ω_i$, there is a point $q\in \cup_k R^{-k}(p_0), k\geq0$ so that the Kobayashi distance $d_{Ω_i}(z_0, q)\leq C.$ If all $Ω_i$ are simply connected, then the result is the same as for polynomials and is treated in an earlier paper. For entire functions $f$, we generally can not have similar results as for rational functions. However, if $f$ has finitely many critical points, then similar results hold.

math.DS↗
arXiv2021

Notes on Short $\mathbb{C}^k$'s

Domains that are increasing union of balls (up to biholomorphism) and on which the Kobayashi metric vanishes identically arise inexorably in complex analysis. In this article we show that in higher dimensions these domains have infinite volume and the Bergman spaces of these domains are trivial. As a consequence they fail to be strictly pseudo-convex at each of their boundary points although these domains are pseudo-convex by definition. These domains can be of different types and one of them is Short $\mathbb{C}^k$'s. In pursuit of identifying the Runge Short $\mathbb{C}^k$'s (up to biholomorphism), we introduce a special class of Short $\mathbb{C}^k$'s, called Loewner Short $\mathbb{C}^k$'s. These are those Short $\mathbb{C}^k$'s which can be exhausted in a continuous manner by a strictly increasing parametrized family of open sets, each of which is biholomrphically equivalent to the unit ball and therefore, they are Runge up to biholomorphism. Although, the question of whether all Short $\mathbb{C}^k$'s are Runge (up to biholomorphism), or whether all Short $\mathbb{C}^k$'s are Loewner remains unsettled, we show that the typical Short $\mathbb{C}^k$'s are Loewner. In the final section, we construct a bunch of non-autonomous basins of attraction, which serve as interesting examples of Short $\mathbb{C}^2$'s.

math.CV↗
arXiv2020

Holomorphic approximation: the legacy of Weierstrass, Runge, Oka-Weil, and Mergelyan

In this paper we survey the theory of holomorphic approximation, from the classical 19th century results of Runge and Weierstrass, continuing with the 20th century work of Oka and Weil, Mergelyan, Vitushkin and others, to the most recent ones on higher dimensional manifolds. The paper includes some new results and applications of this theory, especially to manifold-valued maps.

math.CV↗
arXiv2018

Comparison of invariant metrics and distances on strongly pseudoconvex domains and worm domains

We prove that for a strongly pseudoconvex domain $D\subset\mathbb C^n$, the infinitesimal Carathéodory metric $g_C(z,v)$ and the infinitesimal Kobayashi metric $g_K(z,v)$ coincide if $z$ is sufficiently close to $bD$ and if $v$ is sufficiently close to being tangential to $bD$. Also, we show that every two close points of $D$ sufficiently close to the boundary and whose difference is almost tangential to $bD$ can be joined by a (unique up to reparameterization) complex geodesic of $D$ which is also a holomorphic retract of $D$. The same continues to hold if $D$ is a worm domain, as long as the points are sufficiently close to a strongly pseudoconvex boundary point. We also show that a strongly pseudoconvex boundary point of a worm domain can be globally exposed, this has consequences for the behavior of the squeezing function.

math.CV↗
arXiv2017

Dynamics of transcendental Hénon maps

The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can occur for polynomials, there also exist so-called Baker domains, periodic components where all orbits converge to infinity, as well as wandering domains. In trying to find analogues of these one dimensional results, it is not clear which higher dimensional transcendental maps to consider. In this paper we find inspiration from the extensive work on the dynamics of complex Hénon maps. We introduce the family of transcendental Hénon maps, and study their dynamics, emphasizing the description of Fatou components. We prove that the classification of the recurrent invariant Fatou components is similar to that of polynomial Hénon maps, and we give examples of Baker domains and wandering domains.

math.CV↗
arXiv2016

Exposing boundary points of strongly pseudoconvex subvarieties in complex spaces

We prove that all locally exposable points in a Stein compact in a complex space can be exposed along a given curve to a given real hypersurface. Moreover, the exposing map for a boundary point can be sufficiently close to the identity map outside any fixed neighborhood of the point. We also prove a parametric version of this result for bounded strongly pseudoconvex domains in $\mathbb C^n$. For a bounded strongly pseudoconvex domain in $\mathbb C^n$ and a given boundary point of it, we prove that there is a global coordinate change on the closure of the domain which is arbitrarily close to the identity map with respect to the $C^1$-norm and maps the boundary point to a strongly convex boundary point.

math.CV↗
arXiv2012

The range of holomorphic maps at boundary points

We prove a boundary version of the open mapping theorem for holomorphic maps between strongly pseudoconvex domains. That is, we prove that the local image of a holomorphic map $f:D\to D'$ close to a boundary regular contact point $p\in \de D$ where the Jacobian is bounded from zero along normal non-tangential directions has to eventually contain every cone (and more generally every admissible region) with vertex at $f(p)$.

math.CV↗
arXiv2012

Equilibrium measures on saddle sets of holomorphic maps on P^2

We consider the case of hyperbolic basic sets $Λ$ of saddle type for holomorphic maps $f: \mathbb P^2\mathbb C \to \mathbb P^2\mathbb C$. We study equilibrium measures $μ_ϕ$ associated to a class of Hölder potentials $ϕ$ on $Λ$, and find the measures $μ_ϕ$ of iterates of arbitrary Bowen balls. Estimates for the pointwise dimension $δ_{μ_ϕ}$ of $μ_ϕ$ that involve Lyapunov exponents and a correction term are found, and also a formula for the Hausdorff dimension of $μ_ϕ$ in the case when the preimage counting function is constant on $Λ$. For terminal/minimal saddle sets we prove that an invariant measure $ν$ obtained as a wedge product of two positive closed currents, is in fact the measure of maximal entropy for the \textit{restriction} $f|_Λ$. This allows then to obtain formulas for the measure $ν$ of arbitrary balls, and to give a formula for the pointwise dimension and the Hausdorff dimension of $ν$.

math.DS↗