Searcharxiv⌕ Search

arXiv subjects

Jan Kiwi

Publications and source records attributed to Jan Kiwi.

18 recordsLinked to original sources
arXiv2024

Compactifications and measures for rational maps

We study extensions of the measure of maximal entropy to suitable compactifications of the parameter space and the moduli space of rational maps acting on the Riemann sphere. For parameter space, we consider a space which resolves the discontinuity of the iterate map. We show that the measure of maximal entropy extends continuously to this resolution space. For moduli space, we consider a space which resolves the discontinuity of the iterate map acting on its geometric invariant theory compactification. We show that the measure of maximal entropy, barycentered and modulo rotations, also extends continuously to this resolution space. Thus, answering in the positive a question raised by DeMarco. A main ingredient is a description of limiting dynamics for some sequences.

math.DS↗
arXiv2022

The basin of infinity of tame polynomials

Let $\mathbb{C}_v$ be a characteristic zero algebraically closed field which is complete with respect to a non-Archimedean absolute value. We provide a necessary and sufficient condition for two tame polynomials in $\mathbb{C}_v[z]$ of degree $d \ge 2$ to be analytically conjugate on their basin of infinity. In the space of monic centered polynomials, tame polynomials with all their critical points in the basin of infinity form the tame shift locus. We show that a tame map $f\in\mathbb{C}_v[z]$ is in the closure of the tame shift locus if and only if the Fatou set of $f$ coincides with the basin of infinity.

math.DS↗
arXiv2021

Irreducibility of periodic curves in cubic polynomial moduli space

In the moduli space of complex cubic polynomials with a marked critical point, given any p>=1, we prove that the loci formed by polynomials with the marked critical point periodic of period p is an irreducible curve. Thus answering a question posed by Milnor in the 90's.

math.DS↗
arXiv2020

Indeterminacy loci of iterate maps in moduli space

The moduli space $\mathrm{rat}_d$ of rational maps in one complex variable of degree $d \ge 2$ has a natural compactification by a projective variety $\overline{\mathrm{rat}}_d$ provided by geometric invariant theory. Given $n \ge 2$, the iteration map $Φ_n : \mathrm{rat}_d \to\mathrm{rat}_{d^n}$, defined by $Φ_n: [f] \mapsto [f^n]$, extends to a rational map $Φ_n : \overline{\mathrm{rat}}_d\dashrightarrow \overline{\mathrm{rat}}_{d^n}$. We characterize the elements of $\overline{\mathrm{rat}}_d$ which lie in the indeterminacy locus of $Φ_n$.

math.DS↗
arXiv2019

Irreducibility of the set of cubic polynomials with one periodic critical point

The space of monic centered cubic polynomials with marked critical points is isomorphic to C^2. For each n>0, the locus Sn formed by all polynomials with a specified critical point periodic of exact period n forms an affine algebraic set. We prove that Sn is irreducible, thus giving an affirmative answer to a question posed by Milnor. (This manuscript has been withdrawn)

math.DS↗
arXiv2012

Rescaling limits of complex rational maps

We discuss rescaling limits for sequences of complex rational maps in one variable which approach infinity in parameter space.It is shown that any given sequence of maps of degree $d \ge 2$ has at most $2d-2$ dynamically distinct rescaling limits which are not postcritically finite. For quadratic rational maps, a complete description of the possible rescaling limits is given. These results are obtained employing tools from non-Archimedean dynamics.

math.DS↗
arXiv2012

Combinatorics and topology of straightening maps I: compactness and bijectivity

We study the parameter space structure of degree $d \ge 3$ one complex variable polynomials as dynamical systems acting on $\C$. We introduce and study {\it straightening maps}. These maps are a natural higher degree generalization of the ones introduced by Douady and Hubbard to prove the existence of small copies of the Mandelbrot set inside itself. We establish that straightening maps are always injective and that their image contains all the corresponding hyperbolic systems. Also, we characterize straightening maps with compact domain. Moreover, we give two classes of bijective straightening maps. The first produces an infinite collection of embedded copies of the $(d-1)$-fold product of the Mandelbrot set in the connectedness locus of degree $d \ge 3$. The second produces an infinite collection of full families of quadratic connected filled Julia sets in the cubic connectedness locus, such that each filled Julia set is quasiconformally embedded.

math.DS↗
arXiv2012

Counting Hyperbolic Components

We give formulas for the numbers of type II and type IV hyperbolic components in the space of quadratic rational maps, for all fixed periods of attractive cycles.

math.DS↗
arXiv2011

A non-archimedean Montel's theorem

We prove a version of Montel's theorem for analytic functions over a non-archimedean complete valued field. We propose a definition of normal family in this context, and give applications of our results to the dynamics of non-archimedean entire functions.

math.AG↗
arXiv2009

Cubic Polynomial Maps with Periodic Critical Orbit, Part II: Escape Regions

The parameter space $\mathcal{S}_p$ for monic centered cubic polynomial maps with a marked critical point of period $p$ is a smooth affine algebraic curve whose genus increases rapidly with $p$. Each $\mathcal{S}_p$ consists of a compact connectedness locus together with finitely many escape regions, each of which is biholomorphic to a punctured disk and is characterized by an essentially unique Puiseux series. This note will describe the topology of $\mathcal{S}_p$, and of its smooth compactification, in terms of these escape regions. It concludes with a discussion of the real sub-locus of $\mathcal{S}_p$.

math.DS↗
arXiv2008

The Lyapunov spectrum is not always concave

We characterize one-dimensional compact repellers having nonconcave Lyapunov spectra. For linear maps with two branches we give an explicit condition that characterizes non-concave Lyapunov spectra.

math.DS↗
arXiv2004

Puiseux series polynomial dynamics and iteration of complex cubic polynomials

We study polynomials with coefficients in a field L as dynamical systems where L is any algebraically closed and complete ultrametric field with dense valuation group and characteristic zero residual field. We give a complete description of the dynamical and parameter space of cubic polynomials. In particular we characterize cubic polynomials with compact Julia sets. Also, we prove that any infraconnected connected component of a filled Julia set (of a cubic polynomial) is either a point or eventually periodic. A smallest field S with the above properties is, up to isomorphism, the completion of the field of formal Puiseux series with coefficients in an algebraic closure of Q. We show that some elements of S naturally correspond to the Fourier series of analytic almost periodic functions (in the sense of Bohr) which parametrize (near infinity) the quasiconformal classes of non-renormalizable complex cubic polynomials. Our techniques are based on the ideas introduced by Branner and Hubbard to study complex cubic polynomials.

math.DS↗
arXiv2004

On the Topology of Solenoidal Attractors of the Cylinder

We study the dynamics of skew product endomorphisms acting on the cylinder $\cyl$, of the form $$ \tht \mapsto (\ell θ, \la θ+ τ(θ)), $$ where $ \ell \geq 2$ is an integer, $\la \in (0,1)$ and $τ: \T \to \R$ is a continuous function. We are interested on {\it topological} properties of the global attractor $\Omegalt$ of this map. Given $\ell$ and a Lipschitz function $τ$, we show that the attractor set $\Omegalt$ is homeomorphic to a closed topological annulus for all $\la$ sufficiently close to 1. Moreover, we prove that $\Omegalt$ is a Jordan curve for at most finitely many $\la \in (0,1)$. These results rely on a detailed study of iterated ``cohomological'' equations of the form $τ= \cL_{\la_1} μ_1$, $μ_1 = \cL_{\la_2} μ_2, >...$, where $\cL_\la μ= μ\circ \ml - \la μ$ and $\ml: \T \to \T$ denotes the multiplication by $\ell$ map. We show the following finiteness result: each Lipschitz function $τ$ can be written in a canonical way as, $$ τ= \cL_{\la_1} \circ ... \circ \cL_{\la_m} μ, $$ where $m \ge 0$, $λ_1, ..., λ_m \in (0, 1]$ and the Lipschitz function $μ$ satisfies $μ\neq \cL_\la ρ$ for every continuous function $ρ$ and every $\la \in (0,1]$.

math.DS↗
arXiv1997

Rational rays and critical portraits of complex polynomials

The aim of this work is to describe the equivalence relations in $\Q/\Z$ that arise as the rational lamination of polynomials with all cycles repelling. We also describe where in parameter space one can find a polynomial with all cycles repelling and a given rational lamination. At the same time we derive some consequences that this study has regarding the topology of Julia sets.

math.DS↗