It is well known that a common and useful method to understand behaviors of a dynamical system generated by iteration of a selfmapping is to find a simple invariant structure in its phase space and to describe the dynamics on it. Invariant manifold is one of such structures and, in particular, invariant curve is the main object for 2dimensional systems and easier to be discussed deeply. The existence of real analytic closed invariant curves for 2dimensional areapreserving mappings has been investigated by many authors [1–7]. In this paper, we deal with the existence of analytic invariant curves for a 2dimensional complex mapping T:{\mathbb{C}}^{2}\to {\mathbb{C}}^{2}, (z,w)\mapsto (\zeta ,\omega ), defined by
\{\begin{array}{l}\zeta =az+bw+\varphi (z,w),\\ \omega =cz+dw+\psi (z,w),\end{array}
(1)
where a, b, c, d are complex constants, b\ne 0, adbc\ne 0, and the power series
\varphi (z,w)=\sum _{i+j\ge 2}{\varphi}_{ij}{z}^{i}{w}^{j}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\psi (z,w)=\sum _{i+j\ge 2}{\psi}_{ij}{z}^{i}{w}^{j}
converge in a neighborhood of the origin. Clearly, the mapping T has a fixed point O=(0,0) with the Jacobian matrix
A=DT(0)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)
at O. The characteristic polynomial is
{P}_{A}(\lambda )={\lambda}^{2}(a+d)\lambda +(adbc).
Observe that the function w=f(z) is an invariant curve of T if and only if f satisfies the functional equation
cz+df(z)+\psi (z,f(z))=f[az+bf(z)+\varphi (z,f(z))],\phantom{\rule{1em}{0ex}}z\in \mathbb{C}.
(2)
Since b\ne 0 and the analytic equation
u=az+by+\varphi (z,y)
(3)
can be uniquely solved for y in the way
y=\frac{a}{b}z+\frac{1}{b}u+\mathrm{\Lambda}(z,u),
(4)
where Λ is analytic in a neighborhood of the origin and ord\mathrm{\Lambda}\ge 2. If we define
h(z):=az+bf(z)+\varphi (z,f(z)),
(5)
then by (3) and (4)
f(z)=\frac{a}{b}z+\frac{1}{b}h(z)+\mathrm{\Lambda}(z,h(z)),
(6)
and hence from (2)
h(h(z))(a+d)h(z)+(adbc)z=\mathrm{\Theta}(z,h(z),h(h(z))),
(7)
where the function
\begin{array}{rcl}\mathrm{\Theta}(z,h(z),h(h(z)))& =& bd\mathrm{\Lambda}(z,h(z))b\mathrm{\Lambda}(h(z),h(h(z)))\\ +b\psi (z,\frac{a}{b}z+\frac{1}{b}h(z)+\mathrm{\Lambda}(z,h(z)))\end{array}
and the power series
\mathrm{\Theta}({z}_{0},{z}_{1},{z}_{2})=\sum _{i+j+k\ge 2}{\mathrm{\Theta}}_{i,j,k}{z}_{0}^{i}{z}_{1}^{j}{z}_{2}^{k}
are analytic in a polydisc.
The transformation (it is called the Schröder transformation)
h(z)=g(\alpha {g}^{1}(z))
(8)
with \alpha \in \mathbb{C} for h yields
g\left({\alpha}^{2}z\right)(a+d)g(\alpha z)+(adbc)g(z)=\mathrm{\Theta}(g(z),g(\alpha z),g\left({\alpha}^{2}z\right)),
(9)
a geometric difference equation.
In order to get an analytic solution of (7), we need to find an invertible analytic solution of equation (9) for possible choices of 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
g(z)=z+\sum _{n\ge 2}{\gamma}_{n}{z}^{n}.
(10)
Substituting (10) into (9) we get
and, for n\ge 2,
\begin{array}{rcl}{P}_{A}\left({\alpha}^{n}\right){\gamma}_{n}& =& \sum _{i+j+k\ge 2}{\mathrm{\Theta}}_{i,j,k}{P}_{n,i,j,k}({\gamma}_{2},\dots ,{\gamma}_{n1};\\ \alpha {\gamma}_{2},\dots ,\alpha {\gamma}_{n1};{\alpha}^{2}{\gamma}_{2},\dots ,{\alpha}^{2}{\gamma}_{n1}),\end{array}
(12)
where {P}_{n,i,j,k} is a homogeneous polynomial with positive coefficients in the variables {\gamma}_{2},\dots ,{\gamma}_{n1}; \alpha {\gamma}_{2},\dots ,\alpha {\gamma}_{n1}; {\alpha}^{2}{\gamma}_{2},\dots ,{\alpha}^{2}{\gamma}_{n1}.
Note equation (11), we have
{P}_{A}\left({\alpha}^{n}\right)=({\alpha}^{n}\alpha ){Q}_{n}(\alpha ),
(13)
where {Q}_{n}(\alpha )={\alpha}^{n}+\alpha (a+d). Hence, for all n\ge 2, (12) can be rewritten as
\begin{array}{rcl}({\alpha}^{n}\alpha ){Q}_{n}(\alpha ){\gamma}_{n}& =& \sum _{i+j+k\ge 2}{\mathrm{\Theta}}_{i,j,k}{P}_{n,i,j,k}({\gamma}_{2},\dots ,{\gamma}_{n1};\\ \alpha {\gamma}_{2},\dots ,\alpha {\gamma}_{n1};{\alpha}^{2}{\gamma}_{2},\dots ,{\alpha}^{2}{\gamma}_{n1}).\end{array}
(14)
In this paper, the complex 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 p1 and l\in \mathbb{Z}\mathrm{\setminus}\{0\}.
Observe that 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 socalled smalldivisor 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
\theta ={a}_{0}+{\theta}_{0}={a}_{0}+\frac{1}{{a}_{1}+{\theta}_{1}}=\cdots ,
denoted simply by \theta =[{a}_{0},{a}_{1},\dots ,{a}_{n},\dots ], where {a}_{j}’s and {\theta}_{j}’s are calculated by the algorithm: (a) {a}_{0}=[\theta ], {\theta}_{0}=\{\theta \}, and (b) {a}_{n}=[\frac{1}{{\theta}_{n1}}], {\theta}_{n}=\{\frac{1}{{\theta}_{n1}}\} for all n\ge 1. Define the sequences {({p}_{n})}_{n\in \mathbb{N}} and {({q}_{n})}_{n\in \mathbb{N}} as follows:
\begin{array}{c}{q}_{2}=1,\phantom{\rule{2em}{0ex}}{q}_{1}=0,\phantom{\rule{2em}{0ex}}{q}_{n}={a}_{n}{q}_{n1}+{q}_{n2},\hfill \\ {p}_{2}=0,\phantom{\rule{2em}{0ex}}{p}_{1}=1,\phantom{\rule{2em}{0ex}}{p}_{n}={a}_{n}{p}_{n1}+{p}_{n2}.\hfill \end{array}
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).