Existence of analytic invariant curves for a complex planar mapping near resonance
© Si; licensee Springer. 2014
Received: 10 August 2014
Accepted: 26 November 2014
Published: 8 December 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.
are analytic in a polydisc.
a geometric difference equation.
where is a homogeneous polynomial with positive coefficients in the variables ; ; .
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.
and . According to (15), we have . This proves that the series (10) is an analytic solution of (9) in a neighborhood of the origin.
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)
Let , and define by the condition . Clearly, is nondecreasing. Then we are able to state the following result.
Lemma 1 (Davie’s lemma )
- (a)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.
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.
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.
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.
where Λ is defined in (4) and g is an invertible analytic solution of equation (9).
The proof is complete. □
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).
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