Related papers: Orbits inside Fatou sets

We prove several results on backward orbits of rational functions over number fields. First, we show that if $K$ is a number field, $\phi\in K(x)$ and $\alpha\in K$ then the extension of $K$ generated by the abelian points in the backward…

We generalise a recent example by F. Bracci, J. Raissy and B. Stens{\o}nes to construct automorphisms of $\mathbb{C}^{d}$ admitting an arbitrary finite number of non-recurrent Fatou components, each biholomorphic to…

We give effective bounds for the set quasi-integral points in orbits of non-isotrivial rational maps over function fields under some conditions, generalizing previous work of Hsia and Silverman (2011) for orbits over function fields of…

Orthogonal rational functions (ORF) on the unit circle generalize orthogonal polynomials (poles at infinity) and Laurent polynomials (poles at zero and infinity). In this paper we investigate the properties of and the relation between these…

Let $d\geq2$ be an integer. The set $\mathbf{F}(d)$ of foliations of degree $d$ on the complex projective plane can be identified with a Zariski's open set of a projective space of dimension $d^2+4d+2$ on which $\mathrm{Aut}(\mathbb…

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 investigate the tautness of invariant Fatou components for holomorphic endomorphisms of P^2. Previously, only basins of attraction were known to be taut. We show that two other kinds of recurrent Fatou components are taut. In the first…

Consider an algebraic function like $F(x) = \sqrt{x^3 - 1}$. If $p \in \mathbb{Q}$ is a rational number, how many iterates of $p$ under $F$ can also be rational? The dynamics of algebraic functions may be formalized in the language of…

Let $K$ be a global function field of characteristic $p$ and degree $D$ over $\mathbb F_{p}(t)$. We consider dynamical systems over the projective line $\mathbb P^1(K)$ defined by rational maps with at most one prime of bad reduction. The…

We investigate the trajectory of an arbitrary $(2,1)$-rational $p$-adic dynamical system in a complex $p$-adic field $\C_p$. (i) In the case where there is no fixed point we show that the $p$-adic dynamical system has a 2-periodic cycle…

A transcendental entire function f is called geometrically finite if the intersection of the set of singular values with the Fatou set is compact and the intersection of the postsingular set with the Julia set is finite. (In particular,…

Let $f\colon \mathbb{C} \to \mathbb{C}$ be a transcendental entire map from the Eremenko-Lyubich class $\mathcal{B}$, and let $\zeta$ be an attracting periodic point of period $p$. We prove that the boundaries of components of the…

We study the discrete dynamical system defined on a subset of $R^2$ given by the iterates of the secant method applied to a real polynomial $p$. Each simple real root $\alpha$ of $p$ has associated its basin of attraction $\mathcal…

In the paper we investigate the behavior of trajectory of rational $p$-adic dynamical system in complex $p$-adic filed $\C_p$. It is studied Siegel disks and attractors of such dynamical systems. We show that Siegel disks may either…

We consider a family of $(2,2)$-rational functions given on the set of complex $p$-adic field $\mathcal{C}_p$. Each such function $f$ has the two distinct fixed points $x_1=x_1(f)$, $x_2=x_2(f)$. We study $p$-adic dynamical systems…

Let f be a transcendental map, and let U be an attracting or parabolic basin, or a doubly parabolic Baker domain. Assume U is simply connected. Then, we prove that periodic points are dense in the boundary of U, under certain hypothesis on…

Let f(z) be a rational function of degree at least 2 with coefficients in a number field K, and assume that the second iterate f^2(z) of f(z) is not a polynomial. The second author previously proved that for any b in K, the forward orbit…

We study the radius of absolute monotonicity R of rational functions with numerator and denominator of degree s that approximate the exponential function to order p. Such functions arise in the application of implicit s-stage, order p…

Dynamics on parabolic immediate basins for rational Newton maps of entire functions have been studied. It is proved that every parabolic immediate basin contains invariant accesses to the parabolic fixed point at infinity. Moreover, among…

Let f_n be a sequence of analytic functions in a domain U with a common attracting fixed point z_0. Suppose that f_n converges to f_0 uniformly on each compact subset of U and that z_0 is a Siegel point of f_0. We establish a sufficient…