Related papers: Non-statistical rational maps

This paper is part of a program to understand the parameter spaces of dynamical systems generated by meromorphic functions with finitely many singular values. We give a full description of the parameter space for a specific family based on…

Let $p \geq 2$ be a prime number and let $\mathbb{C}_p$ be the completion of an algebraic closure of the $p$-adic rational field $\mathbb{Q}_p$. Let $f_c(z)$ be a one-parameter family of rational functions of degree $d\geq 2$, where the…

Lifshitz points are multicritical points at which a disordered phase, a homogeneous ordered phase, and a modulated ordered phase meet. Their bulk universality classes are described by natural generalizations of the standard $\phi^4$ model.…

We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We…

We give a combinatorial classification for the class of postcritically fixed Newton maps of polynomials and indicate potential for extensions. As our main tool, we show that for a large class of Newton maps that includes all hyperbolic…

In this paper we discuss the dynamical structure of the rational family $(f_t)$ given by $$f_t(z)=tz^{m}\Big(\frac{1-z}{1+z}\Big)^{n}\quad(m\ge 2,~t\ne 0).$$ Each map $f_t$ has two super-attracting immediate basins and two free critical…

We explore the occurrence of point configurations within non-meager (second category) Baire sets. A celebrated result of Steinhaus asserts that $A+B$ and $A-B$ contain an interval whenever $A$ and $B$ are sets of positive Lebesgue measure…

We prove that given any $\theta_1,\ldots,\theta_{2d-2}\in \R\setminus\Z$, the support of the bifurcation measure of the moduli space of degree $d$ rational maps coincides with the closure of classes of maps having $2d-2$ neutral cycles of…

We prove in this paper that the set of semi-hyperbolic rational maps has Lebesgue measure zero in the space of rational maps of the Riemann sphere for a fixed degree d at least 2. It generalises an earlier result by J. Graczyk and the…

We use nonlinear signal processing techniques, based on artificial neural networks, to construct an empirical mapping from experimental Rayleigh-Benard convection data in the quasiperiodic regime. The data, in the form of a one-parameter…

We develop a bifurcation theory for infinite dimensional systems satisfying abstract hypotheses that are tailored for applications to mean field coupled chaotic maps. Our abstract theory can be applied to many cases, from globally coupled…

Let $M$ be a compact $n$-dimensional Riemanian manifold, End($M$) the set of the endomorphisms of $M$ with the usual $\mathcal{C}^0$ topology and $\phi: M\to\mathbb{R}$ continuous. We prove that there exists a dense subset of $\mathcal{A}$…

We investigate the family of marked Thurston maps that are defined everywhere on the topological sphere $S^2$, potentially excluding at most countable closed set of essential singularities. We show that when an unmarked Thurston map $f$ is…

Renormalizations can be considered as building blocks of complex dynamical systems. This phenomenon has been widely studied for iterations of polynomials of one complex variable. Concerning non-polynomial hyperbolic rational maps, a recent…

Let $X$ be a variety defined over a number field and $f$ be a dominant rational self-map of $X$ of infinite order. We show that $X$ admits many algebraic points which are not preperiodic under $f$. If $f$ were regular and polarized, this…

We obtain an analog of the prime number theorem for a class of branched covering maps on the $2$-sphere $S^2$ called expanding Thurston maps, which are topological models of some non-uniformly expanding rational maps without any smoothness…

We investigate the set of completely positive, trace-nonincreasing linear maps acting on the set M_N of mixed quantum states of size N. Extremal point of this set of maps are characterized and its volume with respect to the Hilbert-Schmidt…

For the family of Double Standard Maps $f_{a,b}=2x+a+\frac{b}{\pi} \sin2\pi x \quad\pmod{1}$ we investigate the structure of the space of parameters $a$ when $b=1$ and when $b\in[0,1)$. In the first case the maps have a critical point, but…

Let (f_\lambda) be a holomorphic family of rational mappings of degree d on the Riemann sphere, with k marked critical points c_1,..., c_k, parameterized by a complex manifold \Lambda. To this data is associated a closed positive current…

A completely stable multicurve of a post-critically finite rational map induces a combinatorial decomposition. The projections of the small Julia sets are immersed within the original Julia set. We prove that two small Julia sets are…