Brjuno Condition Research Articles

Small divisors in discrete local holomorphic dynamics

We give an introduction to the study of local dynamics of iterated holomorphic mappings near a fixed point via local conjugations in one and several complex variables. Starting with the systematic construction of formal conjugations to Poincaré–Dulac normal form in general and formal linearisations in particular, we discuss conditions for convergence of the normalising series in terms of the linear part. The convergence is closely related to the size of denominators that show up in the normalising series and depend only on the linear part of the mapping. Hence, we speak of a small-divisor problem. The central result on the linearisation problem is the Brjuno condition, that ensures holomorphic linearisability. The first part of these notes is a survey of the local dynamics in one variable from the viewpoint of local normalisations. In this case, the picture is fairly complete. In the second part we introduce the additional obstacles to both formal and holomorphic normalisations that emerge from interactions of multiple eigenvalues, such as resonances, that preclude the Brjuno condition in particular. For these cases, we proceed with several generalisations of the Brjuno condition, that allow us to find convergent conjugations to at least partial normalisations. The last part reviews a recent application of such a partial normalisation to illuminate the local dynamics near certain one-resonant fixed points.

Linearization of ultra-differentiable circle flows beyond Brjuno condition

This paper focuses on the vector fields on the quasi-periodically forced circle flow{φ˙=ϱ+f(φ,θ),θ˙=ω:=(1,α). The system given above is the model in [14], where Krikorian-Wang-You-Zhou prove that the system above is C∞ rotations reducible by assuming that f is analytic in (φ,θ)∈T×T2,T=R/Z, the basic frequency ω is not super-Liouvillean and ρf:=ρ(ϱ+f(φ,θ)) is Diophantine with respect to ω (ρf is fibered rotation number of above system). As the application of the reducible result, some fundamental phenomena in mathematical physics such as the linearizable ones and those displaying mode-locking are locally dense for C∞-topology are also given in [14].We, in this work, improve the result in [14] by weakening the regularity of f to be ultra-differentiable in θ∈T2, f is still analytic in φ∈T and ω,ρf satisfy the same hypotheses in [14]. With the hypotheses above, we also prove that the flow above is C∞ rotations reducible, thus the phenomena in mathematical physics given in [14] will also hold in our setting. Our proof, same as the one in [14], is based on a modified KAM (Kolmogorov–Arnold–Moser) theorem but with the problem from analytic topology to ultra-differentiable topology.

Linearization of multi-frequency quasi-periodically forced circle flows beyond Brjuno condition

In this article, we considered the linearization of analytic quasi-periodically forced circle flows. We generalized the rotational linearization of systems with two-dimensional base frequency to systems with any finite dimensional base frequency case. Meanwhile, we relaxed the arithmetical limitations on the base frequencies. Our proof is based on a generalized Kolmogorov–Arnold–Moser (KAM) scheme.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/22/abstr.html

Linearization of Quasiperiodically Forced Circle Flows Beyond Brjuno Condition

We prove that an analytic quasiperiodically forced circle flow with a not super-Liouvillean base frequency and which is close enough to some constant rotation is $C^{\infty}$ rotations reducible, provided its fibered rotation number is Diophantine with respect to the base frequency. As a corollary, we obtain that among such systems, the linearizable ones and those displaying mode-locking are locally dense for the $C^\infty$-topology.

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.

Analytic invariant curves for a second order difference equation modelled from macroeconomics under the Brjuno condition

In this paper, we investigate analytic invariant curves of a planar mapping which derives a second order difference equation modelled from macroeconomics under the Brjuno condition that is weaker than the Diophantine condition, and present analytic invariant curves by constructing uniformly convergent power series.

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 invariant curves for an iterative equation related to dissipative standard map

In this paper, we study existence of invariant curves of an iterative equation urn:x-wiley:mma:media:mma4260:mma4260-math-0001 which is from dissipative standard map. By constructing an invertible analytic solution g(x) of an auxiliary equation of the form urn:x-wiley:mma:media:mma4260:mma4260-math-0002 invertible analytic solutions of the form g(λg − 1(x)) for the original iterative functional equation are obtained. Besides the hyperbolic case 0 < |λ| < 1, we focus on those λ on the unit circle S1, that is, |λ| = 1. We discuss not only those λ at resonance, that is, at a root of the unity, but also those λ near resonance under the Brjuno condition. Copyright © 2016 John Wiley & Sons, Ltd.

Existence of analytic invariant curves for a complex planar mapping near resonance

In this paper a 2-dimensional mapping is investigated in the complex field ℂ for the existence of analytic invariant curves. Employing the method of majorant series, we need to discuss the eigenvalue α of the mapping at a fixed point. Besides the hyperbolic case , we focus on those α on the unit circle , i.e., . We discuss not only those α at resonance, i.e., at a root of the unity, but also those α near resonance under the Brjuno condition.

Analytic Solutions of a Second-Order Functional Differential Equation

In this paper, we study the existence of analytic solutions of a second-order differential equation $$\begin{aligned} \alpha z+\beta x'(z)+\gamma x''(z)=x(az+bx''(z)), \end{aligned}$$ in the complex field \(\mathbb C,\) where \(\alpha , \beta , \gamma , a, b\) are complex numbers. We discuss not only the constant \(\lambda \) at resonance, i.e. at a root of the unity, but also those \(\lambda \) near resonance (near a root of the unity) under the Brjuno condition.

Analytic invariant curves for an iterative equation related to mosquito model

In this paper, we study the existence of analytic invariant curves of a iterative equation urn:x-wiley:1704214:media:mma3009:mma3009-math-0001 which from mosquito model. By constructing an invertible analytic solution g(x) of an auxiliary equation of the form urn:x-wiley:1704214:media:mma3009:mma3009-math-0002 invertible analytic solutions of the form g(αg − 1(x)) for the original iterative functional equation are obtained. Besides the hyperbolic case 0 < | α | < 1, we focus on those α on the unit circle S1, that is, | α | = 1. We discuss not only those α at resonance, that is at a root of the unity, but also those α near resonance under the Brjuno condition. Copyright © 2013 John Wiley & Sons, Ltd.

Analytic invariant curves for an iterative equation related to Pielou's equation

In this paper, we study the existence of analytic invariant curves of an iterative equation which is from Pielou's equation. By reducing the equation with the Schröder transformation to an auxiliary equation, the author discusses not only that the constant at resonance, i.e. at a root of the unity, but also those near resonance under the Brjuno condition.

Local analytic solutions of a more generalized dhombres equation

We study the local analytic solutions f of the functional equation f(ψ(zf(z))) = ϕ (f(z)) for z in some neighborhood of the origin. Whether the solution f vanishes at z=0 or not plays a critical role for local analytic solutions of this equation. In this paper, we obtain results of analytic solutions not only in the case f(0)=0 but also for f(0)≠ 0. When assuming f(0)=0, for technical reasons, we just get the result for f′(0)≠0. Then when assuming f(0)=ω0≠0, ψ′(0)=s≠0, ψ(z) is analytic at z=0 and ϕ(z) is analytic at z=ω0, we give the existence of local analytic solutions f in the case of 0<|sω_0|<1 and the case of |sω0|=1 with 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 constant λ at resonance, that is, at a root of the unity, but also those λ near resonance under the Brjuno condition.

Exact and explicit solutions for a nonlinear extended iterative differential equation

This paper is concerned with a nonlinear iterative functional differential equation x ′( z ) = 1/ x ( p ( z ) + bx ′( z )). By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. We discuss not only in the general case, but also in critical cases, especially for α given in Schröder transformation is a root of the unity. And in case (H4), we dealt with the equation under the Brjuno condition, which is weaker than the Diophantine condition. Moreover, the exact and explicit solution of the original equation has been investigated for the first time. Such equations are important in both applications and the theory of iterations.

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.

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.

Existence of analytic solutions of an iterative functional equation

This paper is concern analytic solutions of an iterative functional equation of the form f ( p ( z ) + q ( f ( z ) ) ) = h ( f ( z ) ) , z ∈ C . Byconstructing a convergent power series solution of an auxiliary equation g ( ϕ ( α 2 z ) - p ( ϕ ( α z ) ) ) = h ( g ( ϕ ( α z ) - p ( ϕ ( z ) ) ) ) , z ∈ C , analytic solutions of the original equation are obtained. We discuss not only these α appeared in the auxiliary equation at the hyperbolic case 0 < ∣ α∣ ≠ 1 and resonance, i.e., at a root of the unity, but also those α near resonance (i.e., near a root of the unity) under Brjuno condition.

Small divisors problem in dynamical systems and analytic solutions of the Shabat equation

This paper is concerned with the existence of exact analytic solutions to the Shabat equation f ′ ( z ) + q 2 f ′ ( q z ) + f 2 ( z ) − q 2 f 2 ( q z ) = μ , z ∈ C , where μ and q are complex parameters. This equation is the simplest self-similar reduction of so-called dressing chain for constructing and analyzing exactly solvable Schrödinger equation. It is so relevant to the study of the q-oscillator algebra in quantum mechanics. The main aim of this paper is to investigate whether the Shabat equation with | q | = 1 has a non-trivial analytic solution f ( z ) under the Brjuno condition.

Exact and explicit analytic solutions of an extended Jabotinsky functional differential equation

In this paper, an extended Jabotinsky-type functional iterative differential equation is investigated in the complex field C for the existence of local analytic solutions. By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. Furthermore, the exact and explicit analytic solution of the original equation is investigated for the first time. We discuss not only the general case, but also critical cases as well, especially for α a unit root, and in case (H4) we deal with the equation under the Brjuno condition, which is weaker than the Diophantine condition. Such equations are important in both applications and the theory of iteration.