Fatou components of elliptic polynomial skew products

We investigate the description of Fatou components for polynomial skew products in two complex variables. The non-existence of wandering domains near a super-attracting invariant fiber was shown in Lilov [Fatou theory in two dimensions. PhD Thesis, University of Michigan, 2004], and the geometrically attracting case was studied in Peters and Vivas [Polynomial skew products with wandering Fatou-disks. Math. Z.283(1–2) (2016), 349–366] and Peters and Smit [Fatou components of attracting skew products. Preprint, 2015, http://arxiv.org/abs/1508.06605]. In Astorg et al [A two-dimensional polynomial mapping with a wandering Fatou component. Ann. of Math. (2), 184 (2016), 263–313] it was proven that wandering domains can exist near a parabolic invariant fiber. In this paper we study the remaining case, namely the dynamics near an elliptic invariant fiber. We prove that the two-dimensional Fatou components near the elliptic invariant fiber correspond exactly to the Fatou components of the restriction to the fiber, under the assumption that the multiplier at the elliptic invariant fiber satisfies the Brjuno condition and that the restriction polynomial has no critical points on the Julia set. We also show the description does not hold when the Brjuno condition is dropped. Our main tool is the construction of expanding metrics on nearby fibers, and one of the key steps in this construction is given by a local description of the dynamics near a parabolic periodic cycle.

The existence of Siegel disks for the Cremona map

This paper is concerned with the existence of Siegel disks of the Cremona map F?(x,y)=(x cos?-(y-x2) sin?, x sin?+(y-x2) cos?) with the parameter ? ? [0, 2?). This problem is reduced to the existence of local invertible analytic solutions to a functional equation with small divisors ?n + ?-n - ? - ?-1. The main aim of this paper is to investigate whether this equation with |?|=1 has such a solution under the Brjuno condition.

Analytic Solutions for a Functional Differential Equation Related to a Traffic Flow Model

We study the existence of analytic solutions of a functional differential equation(z(s)+α)2z'(s)=β(z(s+z(s))-z(s))which comes from traffic flow model. By reducing the equation with the Schröder transformation to an auxiliary equation, the author discusses not only that the constantλat resonance, that is, at a root of the unity, but also thoseλnear resonance under the Brjuno condition.

The Existence of Analytic Invariant Curves for a Planar Map

In this paper, we study the existence of analytic invariant curves for two-dimensional maps in the complex field C. Employing the method of majorant series, we discuss the eigenvalueof the mapping at a fixed point. We discuss not only thoseat resonance, i.e., at a root of the unity but also thosenear resonance under Brjuno condition.

Analytic Solutions of a Nonhomogeneous Iterative Functional Differential Equation

This paper is concerned with an iterative functional differential equation with state-dependent delay As well as in previous works, we reduce this problem with the Schroeder transformation to obtain auxiliary equation. For technical reasons, in previous work the constantgiven in the Schroeder transformation, is required to fulfill that is off the unit circle or lies on the circle with the Diophantine condition. In this paper, we discuss not only thoseat a root of the unity, but also those near resonance under the Brjuno condition.

A renormalization group approach to quasiperiodic motion with Brjuno frequencies

We introduce a renormalization group scheme that applies to vector fields on 𝕋d×ℝm with frequency vectors that satisfy a Brjuno condition. Earlier approaches were restricted to Diophantine frequencies, owing to a limited control of multidimensional continued fractions. We get around this restriction by avoiding the use of a continued-fractions expansion. Our results concerning invariant tori generalize those of [H. Koch and S. Kocić, Renormalization of vector fields and Diophantine invariant tori. Ergod. Th. & Dynam. Sys.28 (2008), 1559–1585] from Diophantine- to Brjuno-type frequency vectors. In particular, each Brjuno vector ω∈ℝd determines an analytic manifold 𝒲 of infinitely renormalizable vector fields, and each vector field on 𝒲 is shown to have an elliptic invariant d-torus with frequencies ω1,ω2,…,ωd.

HERMAN RINGS OF BLASCHKE PRODUCTS OF DEGREE 3

Let Fa,λbe the Blaschke product of the form Fa,λ= λz2((z - a)/(1 - āz)) and α denote an irrational number satisfying the Brjuno condition. Henriksen [1997] showed that for any α there exists a constant a0≧ 3 and a continuous function λ(a) such that Fa,λ(a)possesses an Herman ring and also that modulus M(a) of the Herman ring approaches 0 as a approaches a0. It is remarked that the question whether a0= 3 holds or not is open. According to the idea of Fagella and Geyer [2003] we can show that for a certain set of irrational rotation numbers, a0is strictly larger than 3.