Elliptic Fixed Points with an Invariant Foliation: Some Facts and More Questions

We address the following question: let F : (R, 0) → (R, 0) be an analytic local diffeomorphism defined in the neighborhood of the non resonant elliptic fixed point 0 and let Φ be a formal conjugacy to a normal form N . Supposing F leaves invariant the foliation by circles centered at 0, what is the analytic nature of Φ and N ? 1 Motivation: the two families Aλ,a,d, Bλ,a,d Understanding the normalization of the following examples of local analytic diffeomorphisms of the plane with an elliptic fixed point was the motivation for raising the questions studied in this paper. Preserving the foliation by circles, these examples are radially trivial but angularly subtle; a normalization is a formal change of coordinates which makes the angular behavior trivial. Aλ,a,d and Bλ,a,d are the local maps from (R, 0) to itself respectively defined by { Aλ,a,d(z) = λz(1 + a|z|2d)eπ(z−z̄) Bλ,a,d(z) = λz(1 + a|z|2d)eπ|z| (2i+z−z̄) , where λ = ρe, 0 < ρ ≤ 1 and a ∈ R, a < 0. In polar coordinates z = re : { Aλ,a,d(r, θ) = ( ρr(1 + ar), θ + ω + r sin 2πθ ) , Bλ,a,d(r, θ) = ( ρr(1 + ar), θ + ω + r + r sin 2πθ ) . We shall use the notations { Aλ(z) = Aλ,0,d(z) = λze π(z−z̄), Bλ(z) = Bλ,0,d(z) = λze π|z|(2i+z−z̄) . ∗IMCCE (Observatoire de Paris, PSL Research University, CNRS) †University Paris 7 ‡Department of Mathematics, Capital Normal University, Beijing 100048, China §Academy for Multidisciplinary Studies, Capital Normal University, Beijing 100048, China 1 ar X iv :2 11 1. 07 71 4v 1 [ m at h. D S] 1 5 N ov 2 02 1 The families, parametrized by r, of analytic diffeomorphisms of the circle defined by the angular component of Aλ,a,d and Bλ,a,d are subfamilies of Arnold’s family θ 7→ θ + s+ t sin 2πθ, whose resonant zones (parameter values for which the rotation number is rational, the so-called Arnold’s tongues) are depicted on figure 1. In particular, each rational rotation number corresponds to an interval of values of r ([A, H]). Figure 1 : Families A and B. 2 Formal theory 2.1 Special normal forms Definition 1 Let F0 be the foliation of R by circles centered at 0. A formal diffeomorphism F : (R, 0) → (R, 0) is said to preserve F0 if |F (z)| depends only on |z|. Identifying R with C, this means that it is of the form  F (z) = λz ( 1 + f(|z|) ) e, where λ 6= 0 ∈ C, f(u) = ∑ n≥1 fnu , fn ∈ R, g(z) = ∑ j+k≥1 gjkz j z̄, gkj = ḡjk ∈ C. Blowing up the fixed point, that is using polar coordinates z = re, turns the (formal) diffeomorphism F into a skew-product over the half-line R (which we shall still call F ): F : R × T → R × T, F (r, θ) = ( r ( 1 + f(r) ) , θ + ω + g(r, θ) ) . Hence, iterating F amounts to composing sequences of (formal) circle diffeomorphisms. The eigenvalues of the linear part dF (0) of F are λ and λ̄. The case |λ| < 1 is well understood since Poincaré: F is then locally formally conjugate to dF (0)