Timoteo Carletti
We study the orbit behavior of a germ of an analytic vector field of $(C^n,0)$, $n \geq 2$. We prove that if its linear part is semisimple, non--resonant and verifies a Bruno--like condition, then the origin is effectively stable: stable for finite but exponentially long times.
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We study the Siegel--Schröder center problem on the linearization of analytic germs of diffeomorphisms in several complex variables, in the Gevrey--$s$, $s>0$ category. We introduce a new arithmetical condition of Bruno type on the linear part of the given germ, which ensures the existence of a Gevrey--$s$ formal linearization. We use this fact to prove the effective stability, i.e. stability for finite but long time, of neighborhoods of the origin for the analytic germ.
We study the behavior of diffeomorphisms, contained in the closure $\bar {\A_\a}$ (in the inductive limit topology) of the set $\A_\a$ of real-analytic diffeomorphisms of the torus $\Bbb T^2$, conjugated to the rotation $R_\a:(x,y)\mapsto (x + \a, y)$ by an analytic measure-preserving transformation. We show that for a generic $\a\in [0,1]$, $\bar {\A_\a}$ contains a dense set of uniquely ergodic diffeomorphisms. We also prove that $\bar {\A_\a}$ contains a dense set of diffeomorphisms that are minimal and non-ergodic.
Let G a group of germs of analytic diffeomorphisms in (C^2,0). We find some remarkable properties supposing that G is finite, linearizable, abelian nilpotent and solvable. In particular, if the group is abelian and has a generic dicritic diffeomorphisms, then the group is a subgroup of a 1-parametric group. In addition, we study the topological behaviour of the orbits of a dicritic diffeomorphisms. Last, we find some invariants in order to know when two diffeomorphisms are formally conjugates.
In [H-Y83], Herman and Yoccoz prove that for any given locally analytic (at $z=0$) power series $f(z)=z(λ+\sum_{i=1}^\infty a_iz^i)$ over a complete non-Archimedean field of characteristic $0$ if $|λ|=1$ and $λ$ is not a root of unity, then $f$ is locally linearizable at $z=0$. They ask the same question for power series over fields of positive characteristic. In this paper, we prove that, on opposite, most such power series in this case are more likely to be non-linearizable. More precisely, given a complete non-Archimedean field $\mathcal K$ of positive characteristic and a power series $f(z…
A local Lorentzian Ferrand-Obata theorem for conformal vector fields
研究闭实解析洛伦兹流形上共形向量场，通过改进局部范式等方法，证明其流局部等距或度量处处共形平坦，此为洛伦兹 - 利希纳罗维茨猜想的局部版本。
We provide a complete system of analytic invariants for unfoldings of non-linearizable resonant complex analytic diffeomorphisms as well as its geometrical interpretation. In order to fulfill this goal we develop an extension of the Fatou coordinates with controlled asymptotic behavior in the neighborhood of the fixed points. The classical constructions are based on finding regions where the dynamics of the unfolding is topologically stable. We introduce a concept of infinitesimal stability leading to Fatou coordinates reflecting more faithfully the analytic nature of the unfolding. These impr…
Bruno ideal and the variety of centers for singular germs of vector fields
本文研究了满足Bruno算术条件的解析向量场的Bruno理想，证明其为解析理想，并展示了其在该理想上的可解析化性质及叶形的可解析线性化。
Dimension of self-conformal measures associated to an exponentially separated holomorphic IFS
针对满足四项特定条件的指数分离全纯迭代函数系，研究证明其关联的自共形测度维数及自共形集的豪斯多夫维数均达到自然上界，证明结合了分形测度维数理论与复分析方法。
We establish normal forms for conformal vector fields on pseudo-Riemannian manifolds in the neighborhood of a singularity. For real-analytic Lorentzian manifolds, we show that the vector field is analytically linearizable or the manifold is conformally flat. In either case, the vector field is locally conjugate to a normal form on a model space. For smooth metrics of general signature, we obtain the analogous result under the additional assumption that the differential of the flow at the fixed point is bounded.
In this paper we study the splitting of separatrices phenomenon which arises when one considers a Hamiltonian System of one degree of freedom with a fast periodic or quasiperiodic and meromorphic in the state variables perturbation. The obtained results are different from the previous ones in the literature, which mainly assume algebraic or trigonometric polynomial dependence on the state variables. As a model, we consider the pendulum equation with several meromorphic perturbations and we show the sensitivity of the size of the splitting on the width of the analyticity strip of the perturbati…
We show that in the Gevrey topology, a $d$-torus flow close enough to linear with a unique rotation vector $ω$ is linearizable as long as $ω$ satisfies a Brjuno type diophantine condition. The proof is based on the fast convergence under renormalization of the associated Gevrey vector field. It requires a multidimensional continued fractions expansion of $ω$, and the corresponding characterization of the Brjuno type vectors. This demonstrates that renormalization methods deal very naturally with Gevrey regularity expressed in the decay of Fourier coefficients. In particular, they provide linea…
In this paper we deal with analytic nonautonomous vector fields with a periodic time-dependancy, that we study near an equilibrium point. In a first part, we assume that the linearized system is split in two invariant subspaces E0 and E1. Under light diophantine conditions on the eigenvalues of the linear part, we prove that there is a polynomial change of coordinates in E1 allowing to eliminate up to a finite polynomial order all terms depending only on the coordinate u0 of E0 in the E1 component of the vector field. We moreover show that, optimizing the choice of the degree of the polynomial…
Analytic centralisers of quadratic germs and the Brjuno dichotomy
该研究针对无理参数二次映射，证明其不可解析线性化时解析中心化子平凡，结合Brjuno-Yoccoz定理得到完整二分结果，推广了相关稠密参数结论，还确定了形式交换子收敛集并分类解析迭代根。
Dimension of self-conformal measures associated to an exponentially separated analytic IFS on $\mathbb{R}$
本文将 Hochman 关于指数分离自相似测度的维数结果推广到实解析迭代函数系统，通过把无限维解析映射空间上的卷积约化为有界次数多项式空间上的卷积并建立熵增结果，证明自共形测度维数公式。
On exponential separation of analytic self-conformal sets on the real line
本文研究了实线上指数分离解析IFS的维度特性，证明其在C²拓扑下稠密，并引入双IFS工具，给出了指数分离的充分条件及构造方法。
On exponentially accurate approximation of a near the identity map by an autonomous flow
本文证明了Neishtadt定理的精细化版本，即解析近恒等映射可由自治流的时间一映射以指数精度逼近，并提供了向量场显式表达式和误差界。
In this paper, we give a new construction of resonant normal forms with a small remainder for near-integrable Hamiltonians at a quasi-periodic frequency. The construction is based on the special case of a periodic frequency, a Diophantine result concerning the approximation of a vector by independent periodic vectors and a technique of composition of periodic averaging. It enables us to deal with non-analytic Hamiltonians, and in this first part we will focus on Gevrey Hamiltonians and derive normal forms with an exponentially small remainder. This extends a result which was known for analytic…
On the length and area spectrum of analytic convex domains
本文为解析共振旋转不变曲线上的保面积扭转映射建立了周期作用量差异的指数级小上界，并将其应用于台球问题，改进了关于解析凸区域周期轨迹长度与面积谱的经典光滑情形结果。
Steepness is a geometric property which, together with complex-analyticity, is needed in order to insure stability of a near-integrable hamiltonian system over exponentially long times. Following a strategy developed by Nekhoro-shev, we construct sufficient algebraic conditions for steepness for a given function that involve algebraic equations on its derivatives up to order five. The underlying analysis suggests some interesting considerations on the gener-icity of steepness and represents a first step towards the construction of sufficient conditions for steepness involving the derivatives o…
The method of continuous averaging can be regarded as a combination of the Lie method, where a change of coordinates is constructed as a shift along solutions of a differential equation and the Neishtadt method, well-known in perturbation theory for ODE in the presence of exponentially small effects. This method turns out to be very effective in the analysis of one- and multi-frequency averaging, exponentially small separatrix splitting and in the problem of an inclusion of an analytic diffeomorphism into an analytic flow. We discuss general features of the method as well as the applications.
We prove that generically, both in a topological and measure-theoretical sense, an invariant Lagrangian Diophantine torus of a Hamiltonian system is doubly exponentially stable in the sense that nearby solutions remain close to the torus for an interval of time which is doubly exponentially large with respect to the inverse of the distance to the torus. We also prove that for an arbitrary small perturbation of a generic integrable Hamiltonian system, there is a set of almost full positive Lebesgue measure of KAM tori which are doubly exponentially stable. Our results hold true for real-analyti…
Exponentially accurate Hamiltonian embeddings of symplectic A-stable Runge--Kutta methods for Hamiltonian semilinear evolution equations
针对解析初始数据的半线性哈密顿偏微分方程，证明一类A-稳定辛龙格-库塔时间半离散化可嵌入修正哈密顿流，误差指数小，从而守恒修正哈密顿量至指数小误差。
This is a proof of an asymptotic formula which describes exponentially small splitting of separatrices in a generic analytic family of area-preserving maps near a Hamiltonian saddle-centre bifurcation. As a particular case and in combination with an earlier work on a Stokes constant for the Hénon map (Gelfreich, Sauzin (2001)), it implies exponentially small transversality of separatrices in the area-preserving Hénon family when the multiplicator of the fixed point is close to one.
A conjecture of Arnold, Kozlov and Neishtadt on the exponentially small measure of the non-torus set in analytic systems with two degrees of freedom is discussed.
We consider the dynamics in a neighborhood of an elliptic equilibrium point with a Diophantine frequency of a symplectic real analytic vector field and we prove the following result of effective stability. Generically, both in a topological and measure-theoretical sense, any solution starting sufficiently close to the equilibrium point remains close to it for an interval of time which is doubly exponentially large with respect to the inverse of the distance to the equilibrium point. We actually prove a more general statement assuming the frequency is only non-resonant. This improves previous r…
A geometric approach to exponentially small splitting: Zero-Hopf bifurcations of arbitrary co-dimension
本文提出了一种几何方法，用于研究任意余维数零霍普夫分支中的指数小分裂，通过分析多项式方程的异宿连接和复相空间中的不变流形解析性。
This paper derives for non-linear, time-varying and feedback linearizable systems simple controller designs to achieve specified state-and timedependent complex convergence rates. This approach can be regarded as a general gain-scheduling technique with global exponential stability guarantee. Typical applications include the transonic control of an aircraft with strongly Mach or time-dependent eigenvalues or the state-dependent complex eigenvalue placement of the inverted pendulum. As a generalization of the LTI Luenberger observer a dual observer design technique is derived for a broad set of…
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