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Existence of analytic invariant curves for a complex planar mapping near resonance
Advances in Difference Equations volume 2014, Article number: 308 (2014)
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.
It is well known that a common and useful method to understand behaviors of a dynamical system generated by iteration of a self-mapping 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 2-dimensional systems and easier to be discussed deeply. The existence of real analytic closed invariant curves for 2-dimensional area-preserving mappings has been investigated by many authors [1–7]. In this paper, we deal with the existence of analytic invariant curves for a 2-dimensional complex mapping , , defined by
where a, b, c, d are complex constants, , , and the power series
converge in a neighborhood of the origin. Clearly, the mapping T has a fixed point with the Jacobian matrix
at O. The characteristic polynomial is
Observe that the function is an invariant curve of T if and only if f satisfies the functional equation
Since and the analytic equation
can be uniquely solved for y in the way
where Λ is analytic in a neighborhood of the origin and . If we define
then by (3) and (4)
and hence from (2)
where the function
and the power series
are analytic in a polydisc.
The transformation (it is called the Schröder transformation)
with for h yields
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 and . Therefore, without loss of generality, we can assume that
Substituting (10) into (9) we get
and, for ,
where is a homogeneous polynomial with positive coefficients in the variables ; ; .
Note equation (11), we have
where . Hence, for all , (12) can be rewritten as
In this paper, the complex α in (9) is chosen in and satisfies the following hypotheses:
for some integers with and , and for all and .
Observe that α is off the unit circle in the case of (H1) but on in the rest of the cases. More difficulties are encountered for α on , as mentioned in the so-called small-divisor problem (seen in , p.22 and p.146 and ). In the case where α is a Diophantine number, i.e., there exist constants and such that for all , the number is ‘far’ from all roots of the unity and was considered in different settings [12–14]. In recent work  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 , for a real number θ, we denote by its integer part, and let . Then every irrational number θ has a unique expression of Gauss’s continued fraction
denoted simply by , where ’s and ’s are calculated by the algorithm: (a) , , and (b) , for all . Define the sequences and as follows:
It is easy to show that . Thus, to every we associate, using its convergence, an arithmetical function . We say that θ is a Brjuno number or that it satisfies the Brjuno condition if . The Brjuno condition is weaker than the Diophantine condition. For example, if for all , where is a constant, then 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 and (H1) holds. Then equation (9) has an analytic solution of the form (10) in a neighborhood of the origin.
Proof We first consider the case . Since and , there is such that
Define a new sequence by and
A simple inductive proof shows that for all . If
then is convergent in a polydisc. Furthermore, if we set , the power series converge also in a polydisc. If , then we have
Define the function
for from a neighborhood of , then satisfies
In view of , , and the implicit function theorem, there exists a unique function , analytic in a neighborhood of the origin, such that
and . According to (15), we have . This proves that the series (10) is an analytic solution of (9) in a neighborhood of the origin.
Now we consider the case . In this case, the formal power series (10) satisfies
with and . With obvious notations we have
Note that and , it follows that there is a positive constant such that for all . Then the result in the case is obtained by applying the result in the case . This completes the proof. □
3 Geometric difference equation under (H2)
In this section we discuss the existence of analytic solutions of the geometric difference equation (9) under (H2). In order to introduce Davie’s lemma, we need to recall some facts in  briefly. Let and be the sequence of partial denominators of Gauss’s continued fraction for θ as in the Introduction. As in , let
Let be the set of integers such that either or for some and in , with , one has and divides . For any integer , define
where . We then define the function as follows:
Let , and define by the condition . Clearly, is nondecreasing. Then we are able to state the following result.
Lemma 1 (Davie’s lemma )
Let . Then
there is a universal constant (independent of n and θ) such that
for all and ,
The main result of this section is the following theorem.
Theorem 2 Assume that , and (H2) holds. Then equation (9) has an analytic solution of the form (10) in a neighborhood of the origin.
Proof Since for all , it follows from (14) that
with . Define a sequence by and
Similar to the proof in Theorem 1, using the implicit function theorem, we can prove that the power series is convergent in a neighborhood of the origin. Thus there is a positive constant ϱ such that
Now, we can deduce, by induction, that for , where is defined in Lemma 1. In fact . For a proof by induction, we assume that , . According to Lemma 1, it follows from (17) and (18) that
Note that for some universal constant . Then
This implies that the convergence radius of (10) is at least . 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.
Define a sequence by and
where L is defined in (17) and
Theorem 3 Assume that , and (H3) holds. If for all , then equation (9) has an analytic solution of the form
in a neighborhood of the origin, where is an arbitrary constant satisfying the inequality , and the sequence is defined in (19). Otherwise, if for some , 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 for some natural number l, then (14) does not hold for since . In that case, (9) has no formal solutions.
If , then there are infinitely many choices of corresponding in (14) and the power series forms a family of functions of infinitely many parameters. We can arbitrarily choose such that , . In what follows, we prove that the series has a nonzero radius of convergence. First of all, note that
It follows from (14) that for all , ,
Further, we can show that
In fact, for an inductive proof, we assume that for all . When , we have . On the other hand, when , from (21) we get
implying (22). Moreover, as in the proof of Theorem 1, we can prove that the series converges in a neighborhood of the origin. Thus the series 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 and . Clearly, the inverse is analytic in a neighborhood of the point . Let
which is also analytic in a neighborhood of the origin. From (9), it is easy to see that
The proof is complete. □
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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).
The author declares that he has no competing interests.