Parabolic curves for diffeomorphisms in (C^2,0)

Epsilon-neighborhoods of orbits of parabolic diffeomorphisms and cohomological equations

In this article, we study analyticity properties of (directed) areas of epsilon-neighborhoods of orbits of parabolic germs. The article is motivated by the question of analytic classification using epsilon-neighborhoods of orbits in the simplest formal class. We show that the coefficient in front of epsilon^2 term in the asymptotic expansion in epsilon, which we call the principal part of the area, is a sectorially analytic function of initial point of the orbit. It satisfies a cohomological equation similar to the standard trivialization equation for parabolic diffeomorphisms. We give necessa…

The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point

In this paper we describe the moduli space of germs of generic families of analytic diffeomorphisms which unfold a parabolic fixed point of codimension 1. In [MRR] (and also [R]), it was shown that the Ecalle-Voronin modulus can be unfolded to give a complete modulus for such germs. The modulus is defined on a ramified sector in the canonical perturbation parameter $\eps$. As in the case of the Ecalle-Voronin modulus, the modulus is defined up to a linear scaling depending only on $\eps$. Here, we characterize the moduli space for such unfoldings by finding the compatibility conditions on the …

仅具有抛物型不动点的区间微分同胚的迭代导数的精确二次增长

Exact quadratic growth for the derivatives of iterates of interval diffeomorphisms with only parabolic fixed points

本文证明仅具抛物型不动点的区间微分同胚，若存在非二次切触且单侧拓扑排斥的不动点，则迭代导数最大增长恰为二次，否则为次二次。

Parabolic Bifurcations in Complex Dimension 2

Parabolic bifurcations in one complex dimension demonstrate a wide variety of interesting dynamical phenomena. In this paper we consider parabolic bifurcations of families of diffeomorphisms in two complex dimensions. Specifically we consider a two variable family of diffeomorphisms $F_ε: M\to M$ given locally by $$F_ε(x,y) = (x + x^2 + ε^2+ ..., b_εy+...)$$ where $|b_ε|<1$, and the `$...$' terms involve $x$, $y$ and $ε$.

有限正则性下1维抛物微分同胚芽的留数与共轭

On residues and conjugacies for germs of 1-D parabolic diffeomorphisms in finite regularity

本文研究固定原点的实直线非平坦微分同胚芽的共轭类，基于Takens和Yoccoz工作，证明低正则共轭下留数不变性，并推广Schwarzian导数值在$C^2$抛物共轭下不变而在$C^1$下可变的结果。

在精细曲线图上具有抛物且非真作用的环面微分同胚及其广义旋转集

Torus diffeomorphisms with parabolic and non-proper actions on the fine curve graph and their generalized rotation sets

本研究证明环面Anosov-Katok类的一般元素在精细曲线图上有抛物非真作用，且存在平面上任意点对称紧凸位似类型的广义旋转集。

Characteristic directions of two-dimensional biholomorphisms

We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $\mathbb{C}^2$ there exist either an analytic curve of fixed points tangent to $[v]$ or $k$ parabolic manifolds where all the orbits are tangent to $[v]$, and that at least one of these parabolic manifolds is or contains a parabolic curve.

Fatou flowers and parabolic curves

In this survey we shall collect the main results known up to now (July 2015) regarding possible generalizations to several complex variables of the classical Leau-Fatou flower theorem in holomorphic parabolic dynamics.

Analytic moduli for parabolic Dulac germs

In this paper we give moduli of analytic classification for parabolic Dulac i.e. almost regular germs. Dulac germs appear as first return maps of hyperbolic polycycles. Their moduli are given by a sequence of Ecalle-Voronin-like germs of analytic diffeomorphisms. We state the result in a bigger class of parabolic generalized Dulac germs having power-logarithmic asymptotic expansions.

On the equidistribution of unstable curves for pseudo-Anosov diffeomorphisms of compact surfaces

Weprovethattheasymptoticsofergodicintegralsalonganinvariant foliation of a toral Anosov diffeomorphism, or of a pseudo-Anosov diffeomorphism on a compact orientable surface of higher genus, are determined (up to a logarithmic error) by the action of the diffeomorphism on the cohomology of the surface. As a consequence of our argument and of the results of Giulietti and Liverani [GL] on horospherical averages, toral Anosov diffeomorphisms have no Ruelle resonances in the open interval $(1,e^{h_{top}} )$.

Stable manifolds of two-dimensional biholomorphisms asymptotic to formal curves

Let $F\in\mathrm{Diff}(\mathbb{C}^2,0)$ be a germ of a holomorphic diffeomorphism and let $Γ$ be an invariant formal curve of $F$. Assume that the restricted diffeomorphism $F|_Γ$ is either hyperbolic attracting or rationally neutral non-periodic (these are the conditions that the diffeomorphism $F|_Γ$ should satisfy, if $Γ$ were convergent, in order to have orbits converging to the origin). Then we prove that $F$ has finitely many stable manifolds, either open domains or parabolic curves, consisting of and containing all converging orbits asymptotic to $Γ$. Our results generalize to the case …

A fixed point curve theorem for finite orbits local diffeomorphisms

We study local biholomorphisms with finite orbits in some neighborhood of the origin since they are intimately related to holomorphic foliations with closed leaves. We describe the structure of the set of periodic points in dimension 2. As a consequence we show that given a local biholomorphism $F$, in dimension 2 with finite orbits, there exists an analytic curve passing through the origin and contained in the fixed point set of some non-trivial iterate of $F.$ As an application we obtain that at least one eigenvalue of the linear part of $F$ at the origin is a root of unity. Moreover, we sho…

Normal form à la Moser for diffeomorphisms and generalization of Rüssmann's translated curve theorem to higher dimension

We prove a discrete time analogue of 1967 Moser's normal form of real analytic perturbations of vector fields possessing an invariant, reducible, Diophantine torus; in the case of diffeomorphisms too, the persistence of such an invariant torus is a phenomenon of finite co-dimension. Under convenient non-degeneracy assumptions on the diffeomorphisms under study (torsion property for example), this co-dimension can be reduced. As a by-product we obtain generalizations of Rüssmann's translated curve theorem in any dimension, by a technique of elimination of parameters.

Actions of symplectic homeomorphisms/diffeomorphisms on foliations by curves in dimension 2

The two main results of this paper concern the regularity of the invariant foliation of a C0-integrable symplectic twist diffeomorphisms of the 2-dimensional annulus, namely that $\bullet$ the generating function of such a foliation is C1 ; $\bullet$ the foliation is H{ö}lder with exponent 1/2. We also characterize foliations by graphs that are straightenable via a symplectic homeomorphism and prove that every symplectic homeomorphism that leaves invariant all the leaves of a straightenable foliation has Arnol'd-Liouville coordinates, in which the Dynamics restricted to the leaves is conjugate…

Towards a semilocal study of parabolic invariant curves for fibred holomorphic maps

We introduce the study of the local dynamics around a parabolic indifferent invariant curve for fibred holomorphic maps. As in the classical non-fibred case, we show that petals are the main ingredient. Nevertheless, one expects the properties of the base rotation number should play an important role in the arrangement of the petals. We exhibit examples where the existence and the number of petals depend not just on the complex coordinate of the map, but on the base rotation number. Furthermore, under additional hypothesis on the arithmetics and smoothness of the map, we present a theorem that…

On quasi-invariant curves

Quasi-invariant curves are used in the study of hedgehog dynamics. Denjoy-Yoccoz lemma is the preliminary step for Yoccoz's complex renormalization techniques for the study of linearization of analytic circle diffeomorphisms. We give a geometric interpretation of Denjoy-Yoccoz lemma using the hyperbolic metric that gives a direct construction of quasi-invariant curves without renormalization.

Parabolic arcs of the multicorns: Real-analyticity of Hausdorff dimension, and singularities of $\mathrm{Per}_n(1)$ curves

The boundaries of the hyperbolic components of odd period of the multicorns contain real-analytic arcs consisting of quasi-conformally conjugate parabolic parameters. One of the main results of this paper asserts that the Hausdorff dimension of the Julia sets is a real-analytic function of the parameter along these parabolic arcs. This is achieved by constructing a complex one-dimensional quasiconformal deformation space of the parabolic arcs which are contained in the dynamically defined algebraic curves $\mathrm{Per}_n(1)$ of a suitably complexified family of polynomials. As another applicat…

Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves

We prove that every $C^\infty$-smooth, area preserving diffeomorphism of the closed 2-disk having not more than one periodic point is the uniform limit of periodic $C^\infty$-smooth diffeomorphisms. In particular every smooth irrational pseudo-rotation can be $C^0$-approximated by integrable systems. This partially answers a long standing question of A. Katok regarding zero entropy Hamiltonian systems in low dimensions. Our approach uses pseudoholomorphic curve techniques from symplectic geometry.

Cubic polynomials with a parabolic point

We consider the family of cubic polynomials with a simple parabolic fixed point. We prove that the boundary of the immediate basin of attraction of the parabolic point is a Jordan curve and give a description of the dynamics.

Differentiable invariant manifolds of nilpotent parabolic points

We consider a map $F$ of class $C^r$ with a fixed point of parabolic type whose differential is not diagonalizable and we study the existence and regularity of the invariant manifolds associated with the fixed point using the parameterization method. Concretely, we show that under suitable conditions on the coefficients of $F$, there exist invariant curves of class $C^r$ away from the fixed point, and that they are analytic when $F$ is analytic. The differentiability result is obtained as an application of the fiber contraction theorem. We also provide an algorithm to compute an approximation …

环面的精细曲线图上的抛物等距

Parabolic isometries of the fine curve graph of the torus

本文通过证明环面同胚在精细曲线图上椭圆作用等价于对某非零有理向量有有界偏差，完成了其作用分类，并引入慢旋转集作为证明工具。

Asymptotic lines and parabolic points of plane fields in $\mathbb{R}^3$

In this paper are studied the simplest qualitative properties of asymptotic lines of a plane field in Euclidean space. These lines are the integral curves of the null directions of the normal curvature of the plane field, on the closure of the hyperbolic region, where the Gaussian curvature is negative. When the plane field is completely integrable, these curves coincides with the classical asymptotic lines on surfaces.

近抛物映射的 Buff 形式与不变曲线

Buff forms and invariant curves of near-parabolic maps

本文利用 X. Buff 引入的亚纯 1-形式构建了研究近抛物映射局部动力学的通用框架，并证明了当乘子非切向趋近于单位根时不变曲线的驯顺性与外射线的豪斯多夫连续性。