Smooth linearization of commuting circle diffeomorphisms

On simultaneous linearization of certain commuting nearly integrable diffeomorphisms of the cylinder

Let $\mathcal{F}$ and $\mathcal{K}$ be commuting $C^\infty$ diffeomorphisms of the cylinder $\mathbb{T}\times\mathbb{R}$ that are, respectively, close to $\mathcal{F}_0 (x, y)=(x+ω(y), y)$ and $T_α(x, y)=(x+α, y)$, where $ω(y)$ is non-degenerate and $α$ is Diophantine. Using the KAM iterative scheme for the group action we show that $\mathcal{F}$ and $\mathcal{K}$ are simultaneously $C^\infty$-linearizable if $\mathcal{F}$ has the intersection property (including the exact symplectic maps) and $\mathcal{K}$ satisfies a semi-conjugacy condition. We also provide examples showing necessity of the…

A connectedness result for commuting diffeomorphisms of the interval

Let D^r_+[0,1], r >= 1, denote the group of orientation-preserving C^r diffeomorphisms of [0,1]. We show that any two representations of Z^2 in D^r_+[0,1], r >= 2, are connected by a continuous path of representations of Z^2 in D^1_+[0,1]. We derive this result from the classical works by G. Szekeres and N. Kopell on the C^1 centralizers of the diffeomorphisms of [0,1) which are at least C^2 and fix only 0.

Localizing common fixed points of commuting diffeomorphisms of the plane

We prove that if $G\subset\text{Diff}^{1}(\mathbb{R}^2)$ is an Abelian subgroup generated by a family of commuting diffeomorphisms of the plane, all of which are $C^{1}$-close to the identity in the strong $C^{1}$-topology, and if there exist a point $p\in\mathbb{R}^2$ whose orbit is bounded under the action of $G$, then the elements of $G$ have a common fixed point in the convex hull of $\bar{\mathcal{O}_{p}(G)}$. Here, $\bar{\mathcal{O}_{p}(G)}$ denotes the topological closure of the orbit of $p$ by $G$.

Commuting circle diffeomorphisms with their derivatives having mixed moduli of continuity

Let $d\geq 2$ be an integer and let $ω_1,\cdots ,ω_d$ be moduli of continuity in a specified class which contains the moduli of Hölder continuity. Let $f_k$, $k\in\{1,\cdots,d\}$, be $C^{1+ω_k}$ orientation preserving diffeomorphisms of the circle and $f_1,\cdots, f_d$ commute with each other. We prove that if the rotation numbers of $f_k$'s are independent over the rationals and $ω_1(t)\cdotsω_d(t)=tω(t)$ with $\lim_{t\rightarrow 0^+}ω(t)=0$, then $f_1,\cdots,f_d$ are simultaneously (topologically) conjugate to rigid rotations.

李雅普诺夫谱刚性与随机阿诺索夫微分同胚的同时线性化

Lyapunov spectrum rigidity and simultaneous linearization for random Anosov diffeomorphisms

本文研究了随机阿诺索夫微分同胚的李雅普诺夫谱刚性，证明了在特定条件下系统可同时线性化，并探讨了正矩阵随机游走的正李雅普诺夫指数刚性。

交换圆微分同胚的同时重整化

Simultaneous renormalization of commuting circle diffeomorphisms

本文为交换圆微分同胚对定义重整化算子，在加速Brun算法驱动下，证明对有界型旋转向量指数快速收敛到刚性旋转，并推广到满测度类，同时证明算子拓扑双曲性。

Simultaneous linearization of commuting germs of holomorphic diffeomorphisms

Let f_1,...,f_N be commuting germs of holomorphic diffeomorphisms in C fixing the origin with irrational rationally independent rotation numbers alpha_1,...,alpha_N. We adapt Yoccoz' renormalization of germs to this setting to show that a Brjuno-type condition on simultaneous Diophantine approximability of the rotation numbers is sufficient for simultaneous linearizability of f_1,...,f_N. This generalizes a result of Moser's. In the absence of periodic orbits we show that a weaker arithmetic condition analogous to that of Perez-Marco's for the case of a single germ is sufficent for linearizabi…

Holomorphic linearization of commuting germs of holomorphic maps

Let $f_1, ..., f_h$ be $h\ge 2$ germs of biholomorphisms of $\C^n$ fixing the origin. We investigate the shape a (formal) simultaneous linearization of the given germs can have, and we prove that if $f_1, ..., f_h$ commute and their linear parts are almost simultaneously Jordanizable then they are simultaneously formally linearizable. We next introduce a simultaneous Brjuno-type condition and prove that, in case the linear terms of the germs are diagonalizable, if the germs commutes and our Brjuno-type condition holds, then they are holomorphically simultaneously linerizable. This answers to a…

A Note on Commuting Diffeomorphisms on Surfaces

Let S be a closed surface with nonzero Euler characteristic. We prove the existence of an open neighborhood V of the identity map of S in the C^1-topology with the following property: if G is an abelian subgroup of Diff^1(S) generated by any family of elements in V then the elements of G have common fixed points. This result generalizes a similar result due to Bonatti and announced in his paper "Diffeomorphismes commutants des surfaces et stabilite des fibrations en tores".

Koopman principle eigenfunctions and linearization of diffeomorphisms

This paper considers a nonlinear dynamical system on a complex, finite dimensional Banach space which has an asymptotically stable, hyperbolic fixed point. We investigate the connection between the so-called principle eigenfunctions of the Koopman operator and the existence of a topological conjugacy between the nonlinear dynamics and its linearization in the neighborhood of the fixed point. The principle eigenfunctions generate an algebra of observables for the linear dynamics --- called the principle algebra --- which can be used to generate a sequence of approximate conjugacy maps in the sa…

On a differentiable linearization theorem of Philip Hartman

A linear automorphism of Euclidean space is called bi-circular its eigenvalues lie in the disjoint union of two circles $C_1$ and $C_2$ in the complex plane where the radius of $C_1$ is $r_1$, the radius of $C_2$ is $r_2$, and $0 < r_1 < 1 < r_2$. A well-known theorem of Philip Hartman states that a local $C^{1,1}$ diffeomorphism $T$ of Euclidean space with a fixed point $p$ whose derivative $DT_p$ is bi-circular is $C^{1,β}$ linearizable near $p$. We generalize this result to $C^{1,α}$ diffeomorphisms $T$ where $0 < α< 1$. We also extend the result to local diffeomorphisms in Banach spaces wi…

Constant periodic data and rigidity

In this work we lead with expanding maps of the circle and Anosov diffeomorphisms on $\mathbb{T}^d, d \geq 2.$ We prove that, for these maps, \textit{constant periodic data} imply \textit{same periodic data of these maps and their linearizations}, so in particular we have smooth conjugacy. For expanding maps of the circle and Anosov diffeomorphism on $\mathbb{T}^d, d= 2, 3,$ we have global rigidity. In higher dimensions, $d \geq 4,$ we can establish a result of local rigidity, in several cases. The main tools of this work are celebrated results of rigidity involving same periodic data with lin…

On the conservative pasting lemma

Several perturbation tools are established in the volume preserving setting allowing for the pasting, extension, localized smoothing and local linearization of vector fields. The pasting and local linearization hold in all classes of regularity ranging from $C^{1}$ to $C^{\infty}$ (Hölder included). For diffeomorphisms, a conservative linearized version of Franks lemma is proved in the $C^{r,α}$ ($r\in\mathbb{Z}^+$, $0<α<1$) and $C^{\infty}$ settings, the resulting diffeomorphism having the same regularity as the original one.

线性分式旋转区间交换变换

Linear-fractional rotational interval exchange transformations

本文推广了带断点圆微分同胚的刚性结果，将时间反转对偶性从单个断点扩展到多个断点，引入偶数个线性分式映射的交换集合，并满足特定交换关系。

Pathological center foliation with dimension greater than one

In this paper we are considering partially hyperbolic diffeomorphims of the torus, with $dim(E^c) > 1.$ We prove, under some conditions, that if the all center Lyapunov exponents of the linearization $A,$ of a \mbox{DA-diffeomorphism} $f,$ are positive and the center foliation of $f$ is absolutely continuous, then the sum of the center Lyapunov exponents of $f$ is bounded by the sum of the center Lyapunov exponents of $A.$ After, we construct a $C^1-$open class of volume preserving \mbox{DA-diffeomorphisms}, far from Anosov diffeomorphisms, with non compact pathological two dimensional center …

Exponential Mixing and Smooth Classification of Commuting Expanding Maps

We show that genuinely higher rank expanding actions of abelian semi-groups on compact manifolds are $C^{\infty}$-conjugate to affine actions on infra-nilmanifolds. This is based on the classification of expanding diffeomorphisms up to \holder conjugacy by Gromov and Shub, and is similar to recent work on smooth classification of higher rank Anosov actions on tori and nilmanifolds. To prove regularity of the conjugacy in the higher rank setting, we establish exponential mixing of solenoid actions induced from semi-group actions by nilmanifold endomorphisms, a result of independent interest. We…

Linearization of holomorphic germs with quasi-Brjuno fixed points

Let $f$ be a germ of holomorphic diffeomorphism of $\C^n$ fixing the origin $O$, with $df_O$ diagonalizable. We prove that, under certain arithmetic conditions on the eigenvalues of $df_O$ and some restrictions on the resonances, $f$ is locally holomorphically linearizable if and only if there exists a particular $f$-invariant complex manifold. Most of the classical linearization results can be obtained as corollaries of our result.

Analytic linearization of conformal maps of the annulus

We consider holomorphic maps defined in an annulus around $\mathbb R/\mathbb Z$ in $\mathbb C/\mathbb Z$. E. Risler proved that in a generic analytic family of such maps $f_ζ$ that contains a Brjuno rotation $f_0(z)=z+α$, all maps that are conjugate to this rotation form a codimension-1 analytic submanifold near $f_0$. In this paper, we obtain the Risler's result as a corollary of the following construction. We introduce a renormalization operator on the space of univalent maps in a neighborhood of $\mathbb R/\mathbb Z$. We prove that this operator is hyperbolic, with one unstable direction co…

Joint integrability and spectral rigidity for Anosov diffeomorphisms

Let $f\colon\mathbb{T}^d\to\mathbb{T}^d$ be an Anosov diffeomorphism whose linearization $A\in{\rm GL}(d,\mathbb{Z})$ is irreducible. Assume that $f$ is also absolutely partially hyperbolic where a weak stable subbundle is considered as the center subbundle. We show that if the strong stable and unstable subbundles are jointly integrable, then $f$ is dynamically coherent and all foliations match corresponding linear foliation under the conjugacy to the linearization $A$. Moreover, $f$ admits the finest dominated splitting in weak stable subbundle with dimensions matching those for $A$, and it …

曲面微分同胚不变流形的定量估计

Quantitative Estimates on Invariant Manifolds for Surface Diffeomorphisms

本文通过显式不等式量化二维耗散系统轨道的正则性，并将其与局部动力学的拟线性化联系起来，从而在不依赖不变测度的情况下证明正则轨道具有一致有界几何结构的光滑稳定流形和中心流形，为Hénon-like映射重整化理论奠定技术基础。

无聚束条件的光滑不变叶层与随机动力系统的Belitskii $C^{1}$线性化

Smooth invariant foliations without a bunching condition and Belitskii's $C^{1}$ linearization for random dynamical systems

该研究针对随机动力系统，在无Lyapunov指数聚束条件下证明了$C^{1,β}$稳定与不稳定叶层的存在性，进而得到Belitskii型$C^{1,β}$线性化定理，弱化了经典定理的前提假设。

Pathology and asymmetry: centralizer rigidity for partially hyperbolic diffeomorphisms

We discover a rigidity phenomenon within the volume-preserving partially hyperbolic diffeomorphisms with $1$-dimensional center. In particular, for smooth, ergodic perturbations of certain algebraic systems -- including the discretized geodesic flows over hyperbolic manifolds and certain toral automorphisms with simple spectrum and exactly one eigenvalue on the unit circle, the smooth centralizer is either virtually $\mathbb Z^\ell$ or contains a smooth flow. At the heart of this work are two very different rigidity phenomena. The first was discovered in [2,3] for a class of volume-preserving …

The Lagrange inversion formula on non-Archimedean fields. Non-Analytical Form of Differential and Finite Difference Equations

The classical Lagrange inversion formula is extended to analytic and non--analytic inversion problems on non--Archimedean fields. We give some applications to the field of formal Laurent series in $n$ variables, where the non--analytic inversion formula gives explicit formal solutions of general semilinear differential and $q$--difference equations. We will be interested in linearization problems for germs of diffeomorphisms (Siegel center problem) and vector fields. In addition to analytic results, we give sufficient condition for the linearization to belong to some Classes of ultradifferenti…

Quasisymmetric orbit-flexibility of multicritical circle maps

Two given orbits of a minimal circle homeomorphism $f$ are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with $f$. By a well-known theorem due to Herman and Yoccoz, if $f$ is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. As it follows from the a-priori bounds of Herman and Swiatek, the same holds if $f$ is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if $f$ is a critical circle map whose ro…