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

- Wen Si
^{1}Email author

**2014**:308

https://doi.org/10.1186/1687-1847-2014-308

© Si; licensee Springer. 2014

**Received:**10 August 2014**Accepted:**26 November 2014**Published:**8 December 2014

## Abstract

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 $|\alpha |\ne 1$, we focus on those *α* on the unit circle ${S}^{1}$, *i.e.*, $|\alpha |=1$. 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.

## Keywords

- invariant curves
- geometric difference equation
- majorant series
- Brjuno condition
- resonance

## 1 Introduction

*a*,

*b*,

*c*,

*d*are complex constants, $b\ne 0$, $ad-bc\ne 0$, and the power series

*T*has a fixed point $O=(0,0)$ with the Jacobian matrix

*O*. The characteristic polynomial is

*T*if and only if

*f*satisfies the functional equation

*y*in the way

are analytic in a polydisc.

*h*yields

a geometric difference equation.

*α*. This implies that the desired solution satisfies $g(0)=0$ and ${g}^{\prime}(0)\ne 0$. Therefore, without loss of generality, we can assume that

where ${P}_{n,i,j,k}$ is a homogeneous polynomial with positive coefficients in the variables ${\gamma}_{2},\dots ,{\gamma}_{n-1}$; $\alpha {\gamma}_{2},\dots ,\alpha {\gamma}_{n-1}$; ${\alpha}^{2}{\gamma}_{2},\dots ,{\alpha}^{2}{\gamma}_{n-1}$.

*α*in (9) is chosen in $\sigma (A):=\{\lambda \in \mathbb{C}\mid {P}_{A}(\lambda )=0\}$ and satisfies the following hypotheses:

- (H1)
$0<|\alpha |\ne 1$.

- (H2)
$\alpha ={e}^{2\pi i\theta}$, where $\theta \in \mathbb{R}\mathrm{\setminus}\mathbb{Q}$ is a Brjuno number ([8] and [9]),

*i.e.*, $B(\theta )={\sum}_{k=0}^{\mathrm{\infty}}\frac{log{q}_{k+1}}{{q}_{k}}<\mathrm{\infty}$, where $\{{p}_{k}/{q}_{k}\}$ denotes the sequence of partial fractions of the continued fraction expansion of*θ*which is said to satisfy the*Brjuno condition*. - (H3)
$\alpha ={e}^{2\pi iq/p}$ for some integers $p\in \mathbb{N}$ with $p\ge 2$ and $q\in \mathbb{Z}\mathrm{\setminus}\{0\}$, and $\alpha \ne {e}^{2\pi il/k}$ for all $1\le k\le p-1$ and $l\in \mathbb{Z}\mathrm{\setminus}\{0\}$.

*α*is off the unit circle ${S}^{1}$ in the case of (H1) but on ${S}^{1}$ in the rest of the cases. More difficulties are encountered for

*α*on ${S}^{1}$, as mentioned in the so-called small-divisor problem (seen in [10], p.22 and p.146 and [11]). In the case where

*α*is a Diophantine number,

*i.e.*, there exist constants $\zeta >0$ and $\sigma >0$ such that $|{\alpha}^{n}-1|\ge {\zeta}^{-1}{n}^{-\sigma}$ for all $n\ge 1$, the number $\alpha \in {S}^{1}$ is ‘far’ from all roots of the unity and was considered in different settings [12–14]. In recent work [15] the case of (H3), where

*α*is a root of the unity, was also discussed for a general class of iterative equations. Since then, one has been striving to give a result of analytic solutions for those

*α*‘near’ a root of the unity,

*i.e.*, neither being roots of the unity nor satisfying the Diophantine condition. The Brjuno condition in (H2) provides such a chance for us. As stated in [16], for a real number

*θ*, we denote by $[\theta ]$ its integer part, and let $\{\theta \}=\theta -[\theta ]$. Then every irrational number

*θ*has a unique expression of Gauss’s continued fraction

It is easy to show that ${p}_{n}/{q}_{n}=[{a}_{0},{a}_{1},\dots ,{a}_{n}]$. Thus, to every $\theta \in \mathbb{R}\mathrm{\setminus}\mathbb{Q}$ we associate, using its convergence, an arithmetical function $B(\theta )={\sum}_{n\ge 0}\frac{log{q}_{n+1}}{{q}_{n}}$. We say that *θ* is a Brjuno number or that it satisfies the Brjuno condition if $B(\theta )<+\mathrm{\infty}$. The Brjuno condition is weaker than the Diophantine condition. For example, if ${a}_{n+1}\le c{e}^{{a}_{n}}$ for all $n\ge 0$, where $c>0$ is a constant, then $\theta =[{a}_{0},{a}_{1},\dots ,{a}_{n},\dots ]$ is a Brjuno number but is not a Diophantine number. So, the case (H2) contains both a Diophantine condition and a condition which expresses that *α* is near resonance.

In this paper, we consider the Brjuno condition instead of the Diophantine one. We discuss not only the cases (H1) and (H3) but also (H2) for analytic invariant curves of the mapping *T* defined in (1).

## 2 Geometric difference equation under (H1)

**Theorem 1** *Assume that* $\alpha \in \sigma (A)$ *and* (H1) *holds*. *Then equation* (9) *has an analytic solution* $g(z)$ *of the form* (10) *in a neighborhood of the origin*.

*Proof*We first consider the case $0<|\alpha |<1$. Since $ad-bc\ne 0$ and ${lim}_{n\to \mathrm{\infty}}{\alpha}^{n}=0$, there is ${\mathrm{\Theta}}_{0}>0$ such that

and $F(z,\mathrm{\Phi}(z))=0$. According to (15), we have $\beta (z)=\mathrm{\Phi}(z)$. This proves that the series (10) is an analytic solution of (9) in a neighborhood of the origin.

Note that $|\mu |<1$ and $ad-bc\ne 0$, it follows that there is a positive constant ${\mathrm{\Theta}}_{0}$ such that $|{P}_{A}^{\ast}({\mu}^{n})|\ge {\mathrm{\Theta}}_{0}$ for all $n\ge 2$. Then the result in the case $|\alpha |>1$ is obtained by applying the result in the case $|\alpha |<1$. This completes the proof. □

## 3 Geometric difference equation under (H2)

*θ*as in the Introduction. As in [16], let

Let ${g}_{k}(n):=max({h}_{k}(n),[\frac{n}{{q}_{k}}])$, and define $k(n)$ by the condition ${q}_{k(n)}\le n\le {q}_{k(n)+1}$. Clearly, $k(n)$ is nondecreasing. Then we are able to state the following result.

**Lemma 1** (Davie’s lemma [17])

*Let*$K(n)=nlog2+{\sum}_{j=0}^{k(n)}{g}_{j}(n)log(2{q}_{j+1})$.

*Then*

- (a)
*there is a universal constant*$\gamma >0$ (*independent of**n**and**θ*)*such that*$K(n)\le n(\sum _{j=0}^{k(n)}\frac{log{q}_{j+1}}{{q}_{j}}+\gamma ),$ - (b)
$K({n}_{1})+K({n}_{2})\le K({n}_{1}+{n}_{2})$

*for all*${n}_{1}$*and*${n}_{2}$, - (c)
$-log|{\alpha}^{n}-1|\le K(n)-K(n-1)$.

The main result of this section is the following theorem.

**Theorem 2** *Assume that* $\alpha \in \sigma (A)$, $|a+d|>2$ *and* (H2) *holds*. *Then equation* (9) *has an analytic solution* $g(z)$ *of the form* (10) *in a neighborhood of the origin*.

*Proof*Since $|a+d|>2$ for all $n\ge 2$, it follows from (14) that

*ϱ*such that

as required.

This implies that the convergence radius of (10) is at least ${(\varrho {e}^{B(\theta )+\gamma})}^{-1}$. This completes the proof. □

## 4 Geometric difference equation under (H3)

The next theorem is devoted to the case of (H3), where *α* is not only on the unit circle in ℂ but also a root of the unity. In this case neither the Diophantine condition nor the Brjuno condition is satisfied.

*L*is defined in (17) and

**Theorem 3**

*Assume that*$\alpha \in \sigma (A)$, $|a+d|>2$

*and*(H3)

*holds*.

*If*$\mathrm{\Omega}(lp+1,\alpha )=0$

*for all*$l\in \mathbb{N}=\{1,2,\dots \}$,

*then equation*(9)

*has an analytic solution of the form*

*in a neighborhood of the origin*, *where* ${\zeta}_{lp+1}$ *is an arbitrary constant satisfying the inequality* $|{\gamma}_{lp+1}|\le {B}_{lp+1}$, *and the sequence* ${\{{B}_{n}\}}_{n=1}^{\mathrm{\infty}}$ *is defined in* (19). *Otherwise*, *if* $\mathrm{\Omega}(lp+1,\alpha )\ne 0$ *for some* $l=1,2,\dots $ , *then equation* (9) *has no analytic solution in any neighborhood of the origin*.

*Proof* As in the proof of Theorem 1, we seek for a power series solution of (9) of the form (10). Obviously, (11)-(14) hold again. If $\mathrm{\Omega}(lp+1,\alpha )\ne 0$ for some natural number *l*, then (14) does not hold for $n=lp+1$ since ${\alpha}^{lp}-1=0$. In that case, (9) has no formal solutions.

implying (22). Moreover, as in the proof of Theorem 1, we can prove that the series ${\sum}_{n=1}^{\mathrm{\infty}}{B}_{n}{z}^{n}$ converges in a neighborhood of the origin. Thus the series $z+{\sum}_{n\ge 2}{\gamma}_{n}{z}^{n}$ has a nonzero radius of convergence. This completes the proof. □

## 5 Analyticity of invariant curves

In this section, we will state and prove our main results.

**Theorem 4**

*Suppose that one of the conditions in Theorems*1-3

*is fulfilled*.

*Then equation*(2)

*has a solution of the form*

*where* Λ *is defined in* (4) *and* *g* *is an invertible analytic solution of equation* (9).

*Proof*By Theorems 1-3, we can find an analytic solution

*g*of the geometric difference equation (9) in the form of (10) such that $g(0)=0$ and ${g}^{\prime}(0)=\eta \ne 0$. Clearly, the inverse ${g}^{-1}$ is analytic in a neighborhood of the point $g(0)=0$. Let

The proof is complete. □

## Declarations

### Acknowledgements

The author would like to thank the referees for their valuable comments and suggestions which have helped to improve the quality of this paper. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11171185, 10871117).

## Authors’ Affiliations

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