Real Analytic Diffeomorphisms Research Articles

Analytic weakly mixing diffeomorphisms on odd dimensional spheres

We present an approximation by conjugation scheme to obtain real-analytic diffeomorphisms of odd dimensional spheres that are weakly mixing with respect to the volume.

Real-analytic realization of Uniform Circular Systems and some applications

Recently Matthew Foreman and Benjamin Weiss showed in a series of papers that smooth ergodic diffeomorphisms of certain compact manifolds are unclassifiable up to measure-isomorphism. In this paper we show that the uniform circular systems used in the work of Foreman—Weiss admit real-analytic realizations on the two-dimensional torus. As a consequence we obtain the same anti-classification result for real-analytic ergodic diffeomorphisms on the torus. In another application we show the existence of an uncountable family of pairwise non-Kakutani equivalent real-analytic diffeomorphisms on the torus.

On the divergence of Birkhoff Normal Forms

It is well known that a real analytic symplectic diffeomorphism of the 2d-dimensional disk (dgeq 1) admitting the origin as a non-resonant elliptic fixed point can be formally conjugated to its Birkhoff Normal Form, a formal power series defining a formal integrable symplectic diffeomorphism at the origin. We prove in this paper that this Birkhoff Normal Form is in general divergent. This solves, in any dimension, the question of determining which of the two alternatives of Pérez-Marco’s theorem (Ann. Math. (2) 157:557–574, 2003) is true and answers a question by H. Eliasson. Our result is a consequence of the fact that when d=1 the convergence of the formal object that is the BNF has strong dynamical consequences on the Lebesgue measure of the set of invariant circles in arbitrarily small neighborhoods of the origin. Our proof, as well as our results, extend to the case of real analytic diffeomorphisms of the annulus admitting a Diophantine invariant torus.

Real-analytic AbC constructions on the torus

In this article we demonstrate a way to extend the AbC (approximation by conjugation) method invented by Anosov and Katok from the smooth category to the category of real-analytic diffeomorphisms on the torus. We present a general framework for such constructions and prove several results. In particular, we construct minimal but not uniquely ergodic diffeomorphisms and non-standard real-analytic realizations of toral translations.

On the ergodic theory of free group actions by real-analytic circle diffeomorphisms

We consider finitely generated groups of real-analytic circle diffeomorphisms. We show that if such a group admits an exceptional minimal set (i.e., a minimal invariant Cantor set), then its Lebesgue measure is zero; moreover, there are only finitely many orbits of connected components of its complement. For the case of minimal actions, we show that if the underlying group is (algebraically) free, then the action is ergodic with respect to the Lebesgue measure. This provides first answers to questions due to E. Ghys, G. Hector and D. Sullivan.

Non-standard real-analytic realizations of some rotations of the circle

We extend some aspects of the smooth approximation by conjugation method to the real-analytic set-up, and create examples of zero entropy, uniquely ergodic, real-analytic diffeomorphisms of the two-dimensional torus that are metrically isomorphic to some (Liouvillian) irrational rotations of the circle.

The Lie group of real analytic diffeomorphisms is not real analytic

We construct an infinite dimensional real analytic manifold structure for the space of real analytic mappings from a compact manifold to a locally convex manifold. Here a map is real analytic if it extends to a holomorphic map on some neighbourhood of the complexification of its domain. As is well known the construction turns the group of real analytic diffeomorphisms into a smooth locally convex Lie group. We prove then that the diffeomorphism group is regular in the sense of Milnor. In the inequivalent "convenient setting of calculus" the real analytic diffeomorphisms even form a real analytic Lie group. However, we prove that the Lie group structure on the group of real analytic diffeomorphisms is in general not real analytic in our sense.

Analytic uniquely ergodic volume preserving maps on odd spheres

We construct examples of volume-preserving uniquely ergodic (and hence minimal) real-analytic diffeomorphisms on odd-dimemsional spheres.

Second order ODEs under area-preserving maps

We study the geometry of real analytic second order ODEs under the local real analytic diffeomorphism of \(\mathbb {R}^2\) which are area preserving, through the method of Cartan. We obtain a subdivision into three “parts”. The first one is the most symmetric case. It is characterized by the vanishing of an area-preserving relative invariant namely \(f_y+\dfrac{2}{9}f_{y^{\prime }}^{2}-\dfrac{1}{3}\mathfrak {D}(f_{y^{\prime }})\). In this situation we associate a local affine normal Cartan connection on the first jet \(J^{1}(\mathbb {R},\mathbb {R})\) space whose curvature contains all the area-preserving relative differential invariants, to any second order ODE under study. The second case which includes all the Painleve transcendents is given by the ODEs for which \(f_y+\dfrac{2}{9}f_{y^{\prime }}^{2}-\dfrac{1}{3}\mathfrak {D}(f_{y^{\prime }})\not \equiv 0\). In the latter case we give all necessary steps in order to obtain an \(e\)-structure on \(J^{1}(\mathbb {R},\mathbb {R})\) for a generic second order ODE equation of that type. Finally we give the method to reduce to an \(e\)-structure on \(J^{1}\) when \(f_{y^\prime y^\prime y^\prime y^\prime }\not \equiv 0\).

Characterization of the generic unfolding of a weak focus

In this paper we prove that the orbital class of a generic real analytic family unfolding a weak focus is determined by the conjugacy class of its Poincaré monodromy and vice versa. We solve the embedding problem by means of quasiconformal surgery on the formal normal form. The surgery yields an integrable abstract almost complex 2-manifold equipped with an elliptic foliation. The monodromy of the latter coincides with the second iterate of a germ of prescribed family of real analytic diffeomorphisms undergoing a flip bifurcation.

Examples of C 1-smoothly conjugate diffeomorphisms of the circle with break that are not C 1+γ -smoothly conjugate

We prove the existence of two real-analytic diffeomorphisms of the circle with break of the same size and an irrational rotation number of semibounded type that are not C 1+γ -smoothly conjugate for any γ > 0. In this way, we show that the previous result concerning the C 1-smoothness of conjugacy for these mappings is the exact estimate of smoothness for this conjugacy.

The real analytic Feigenbaum–Coullet–Tresser attractor in the disc

We consider a real analytic diffeomorphism ψ0 on an n-dimensional disc 𝒟, n ≥ 2, exhibiting a Feigenbaum–Coullet–Tresser (FCT) attractor. We assume that in the C ω(𝒟) topology it is far from the standard FCT map φ0 fixed by the double renormalization. We prove that ψ0 persists along a codimension-one manifold ℳ ⊂ C ω(𝒟), and that it is the bifurcating map along any one-parameter family in C ω(𝒟) transversal to ℳ, from diffeomorphisms exhibiting sinks to those which exhibit chaos, filling a gap in the usually accepted proof of this assertion. The main tool in the proofs is a theorem of functional analysis, which we state and prove in this article, characterizing the existence of codimension-one submanifolds in any abstract functional Banach space.

Holomorphic versus algebraic equivalence for deformations of real-algebraic Cauchy–Riemann manifolds

Since Poincare’s celebrated paper [19] published in 1907, there has been a growing literature concerned with the equivalence problem for real submanifolds in complex space (see e.g., [4, 6, 7, 11, 13, 14, 22] for some recent works as well as the references therein). One interesting phenomenon, observed by Webster for biholomorphisms of Levi nondegenerate hypersurfaces [23], is that the biholomorphic equivalence of some types of real-algebraic submanifolds of a complex space implies their algebraic equivalence. In this paper, we show that this very phenomenon holds for algebraic deformations of germs of minimal holomorphically nondegenerate realalgebraic CR submanifolds in complex space. Let us recall that a germ of a real-algebraic CR submanifold (M,p) ⊂ (Cn, p) is minimal if there exists no proper CR submanifold N ⊂ M through p of the same CR dimension as M . It is holomorphically nondegenerate if there exists no nontrivial holomorphic vector field tangent to M near p (see [21]). An algebraic deformation of (M,p) is a real-algebraic family of germs at p of real-algebraic CR submanifolds (Ms, p)s∈Rk in Cn, defined for s ∈ Rk near 0, such that M0 = M . We say that two such deformations (Ms, p)s∈Rk and (Nt, p′)t∈Rk are biholomorphically equivalent if there exists a germ of a realanalytic diffeomorphism φ : (Rk, 0) → (Rk, 0) and a holomorphic submersion B : (Cz × Cu, (p, 0)) → (Cn, p′) such that z → B(z, s) is a biholomorphism sending (Ms, p) to (Nφ(s), p′) for all s ∈ Rk close to 0. We shall say that such

On the group of real analytic diffeomorphisms

Le groupe des difféomorphismes analytiques réels d’une variété analytique réelle est un groupe riche. Il est dense dans le groupe des difféomorphismes lisses. Herman a montré que, pour le tore de dimension n, sa composante connexe de l’identité est un groupe simple. Pour les variétés U(1) fibrées, pour les variétés admettant une action semi-libre spéciale de U(1), et pour les variétés de dimension 2 ou 3 admettant une action non-triviale de U(1), on montre que la composante de l’identité du groupe des difféomorphismes analytiques réels est un groupe parfait.

Groups of real analytic diffeomorphisms of the circle with a finite image under the rotation number function

We consider groups of orientation-preserving real analytic diffeomorphisms of the circle which have a finite image under the rotation number function. We show that if such a group is nondiscrete with respect to the C 1 -topology then it has a finite orbit. As a corollary, we show that if such a group has no finite orbit then each of its subgroups contains either a cyclic subgroup of finite index or a nonabelian free subgroup.

Analytic invariants associated with a parabolic fixed point in 2

It is well known that in a small neighbourhood of a parabolic fixed point a real-analytic diffeomorphism of (ℝ2,0) embeds in a smooth autonomous flow. In this paper we show that the complex-analytic situation is completely different and a generic diffeomorphism cannot be embedded in an analytic flow in a neighbourhood of its parabolic fixed point. We study two analytic invariants with respect to local analytic changes of coordinates. One of the invariants was introduced earlier by one of the authors. These invariants vanish for time-one maps of analytic flows. We show that one of the invariants does not vanish on an open dense subset. A complete analytic classification of the maps with a parabolic fixed point in ℂ2 is not available at the present time.

On the symplectic phase space of K\lowercase{d}V

AbstractWe prove that the Birkhoff map Ω for KdV constructed on H −10 (T) canbe interpolated between H −10 (T) and L 2 (T). In particular, the symplecticphase space H 1/20 (T) can be described in terms of Birkhoff coordinates. 1 Introduction In [12] it is shown that the Birkhoff map for the Korteweg - de Vries equation(KdV), on the circle T := R/Z, introduced and studied in detail in [9, 6] can beanalytically extended to an analytic diffeomorphismΩ : H −10 (T) → h −1/2 from the Sobolev space of distributions H −10 (T) (dual of H 10 (T)) to the Hilbertspace of sequences h −1/2 where for any α ∈ R,h α := {z = (x k ,y k ) k≥1 | kzk α < ∞},withkzk α :=X k≥1 k 2α (x 2k +y 2k ) 1/2 .In this paper we show that Ω can be interpolated between H −10 (T) and L 20 (T).Theorem 1. For any −1 ≤ α ≤ 0 , Ω| H α0 (T) : H α0 (T) → h α+1/2 is a real analytic diffeomorphism. As an application of Theorem 1 we characterize the regularity of a potentialq ∈ H −1 (T) in terms of the decay of the gap lengths (γ k

Unitary Representations and Coadjoint Orbits for a Group of Germs of Real Analytic Diffeomorphisms

The method of coadjoint orbits is developed for the group of real analytic germs of diffeomorphisms φ with φ(0) = 0 and φ′(0) = 1. The form of all infinite dimensional coadjoint orbits is described. Classes U of unitary representations are constructed. In the case n = 2 these representations are related to coadjoint orbits.

Extremal mappings of finite distortion

The theory of mappings of finite distortion has arisen out of a need to extend the ideas and applications of the classical theory of quasiconformal mappings to the degenerate elliptic setting where one finds concrete applications in non-linear elasticity and the calculus of variations. In this paper we initiate the study of extremal problems for mappings with finite distortion and extend the theory of extremal quasiconformal mappings by considering integral averages of the distortion function instead of its supremum norm. For instance, we show the following. Suppose that fo is a homeomorphism of the circle with f o − 1 ∈ W 1 / 2 , 2 . Then there is a unique extremal extension to the disk which is a real analytic diffeomorphism with non-vanishing Jacobian determinant. The condition on fo is sharp. Classically the mapping fo is assumed to be quasisymmetric. Then there is an extremal quasiconformal mapping with boundary values fo, but it is not always unique and it is seldom smooth. Indeed, even when fo is quasisymmetric, the L1-minimiser for the distortion function will almost never be quasiconformal. We further find that there are many new and unexpected phenomena concerning existence, uniqueness and regularity for these extremal problems where the functionals are polyconvex but typically not convex. These seem to differ markedly from phenomena observed when studying multi-well type functionals. 2000 Mathematics Subject Classification 30C60, 35J15, 35J70.

A Complete Classification for Pairs of Real Analytic Curves in the Complex Plane with Tangential Intersection

We study the group structure of centralizers of a holomorphic mapping that is tangent to the identity. We give a complete classification for pairs of real analytic curves intersecting tangentially via the Ecalle--Voronin theory. We also classify pairs of one-dimensional holomorphic involutions that have a high-order tangency, real analytic mappings on the real line, and exceptional non-Abelian subgroups of Cerveau and Moussu. Consequently, we prove the existence of holomorphic mappings z ? z + O(2) that are reversible by a formal antiholomorphic mapping, but not by any (convergent) antiholomorphic one. We also prove the existence of local real analytic mappings on the real line that are not equivalent with respect to any local real analytic diffeomorphism and yet whose complexifications are holomorphically equivalent.