A limit theorem for torus translation

Symbolic coding of linear complexity for generic translations of the torus, using continued fractions

In this paper, we prove that almost every translation of $\mathbb{T}^2$ admits a symbolic coding which has linear complexity $2n+1$. The partitions are constructed with Rauzy fractals associated with sequences of substitutions, which are produced by a particular extended continued fraction algorithm in projective dimension $2$. More generally, in dimension $d\geq 1$, we study extended measured continued fraction algorithms, which associate to each direction a subshift generated by substitutions, called $S$-adic subshift. We give some conditions which imply the existence, for almost every direc…

Cibles rétrécissantes de rayon $n^{-\frac{1}{d}}$ : propriété du logarithme

A translation on the d-dimensional torus $\mathbb{T}^d$ has the logarithm property if the Shrinking Target Property holds for the sequence of balls with radius $n^{-\frac{1}{d}}$. On $\mathbb{T}^1$ every irrational translations has this property. In higher dimension, we give criterions based upon an unusual Diophantine type. In the two dimensional torus, we construct counter examples to the logarithm with translations vectors of arbitrary small Diophantine type and example of logarithm property with Liouvillian vector

A homological condition for a dynamical and illuminatory classification of torus branched coverings

We prove that, for translation surfaces whose homology is generated by the periodic orbits, the notions of - finite blocking property - pure periodicity - torus branched covering are equivalent. In particular, we prove this equivalence for convex surfaces and on a dense open subset of full measure on every normalized stratum.

科尔莫戈罗夫不变 torus 定理用于弱相互作用粒子 I：全维 tori

Kolmogorov invariant torus theorem for weakly interacting particles I: Full dimensional tori

本文提出一个抽象KAM定理，用于研究无限多相互作用粒子系统，构建全维KAM tori以应对长程相互作用。

Dynamics of piecewise translation maps

In this paper, we introduce a generalized piecewise translation map on the Euclidean space. We provide a special case when this map is always of finite type. For a finite type map in this case, we form conjectures on the semi-continuity of the attractor. Moreover, we also provide some conjectures and pictures on piecewise translations on a two dimensional torus.

Normal forms for perturbations of systems possessing a Diophantine invariant torus

In 1967 Moser proved the existence of a normal form for real analytic perturbations of vector fields possessing a reducible Diophantine invariant quasi-periodic torus. In this paper we present a proof of existence of this normal form based on an abstract inverse function theorem in analytic class. The given geometrization of the proof can be opportunely adapted accordingly to the specificity of systems under study. In this more conceptual frame, it becomes natural to show the existence of new remarkable normal forms, and provide several translated-torus theorems or twisted-torus theorems for s…

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 nonstandard real-analytic realizations of toral translations.

KAM Theorems for Multi-scale Torus

In present paper, from the viewpoint of physical intuition we introduce a Hamiltonian system with multiscale rotation, which describes many systems, for example, the forced pendulum with fast rotation, weakly coupled $N$-oscillators with quasiperiodic force and so on. We study the persistence of invariant tori for this Hamiltonian system, and establish some KAM type results including the isoenergetic type. As consequences, we can show that Boltzmann's ergodicity hypothesis is also not true for this Hamiltonian system.

Local Rigidity for Simultaneous Diophantine Translations on Tori of Arbitrary Dimension

We show that a smooth sufficiently small perturbation of a $\mathbb Z^m$ action on the torus by simultaneously Diophantine translations, is smoothly conjugate to the unperturbed action under a natural condition on the rotation sets. This generalizes recent result of Karaliolios [4] of the action generators to higher rank abelian actions, and the result of Moser [8] to higher dimensional tori.

No temporal distributional limit theorem for a.e. irrational translation

Bromberg and Ulcigrai constructed piecewise smooth functions f on the torus such that the set of angles alpha for which the Birkhoff sums of f with respect to the irrational translation by alpha satisfies a temporal distributional limit theorem along the orbit of a.e. x has Hausdorff dimension one. We show that the Lebesgue measure of this set of angles is equal to zero.

关于二维环面上的极小同胚反分类定理

An anti-classification theorem for minimal homeomorphisms on the torus

本文证明了二维环面上的极小同胚无法通过可数结构进行分类。

布劳威尔平面平移定理的一个推广

A generalization of the Brouwer plane translation theorem

该研究通过证明含拓扑链回归点的平面保向同胚必有不动点，推广了Brouwer平面平移定理。

Free curves and periodic points for torus homeomorphisms

We study the relationship between free curves and periodic points for torus homeomorphisms in the homotopy class of the identity. By free curve we mean a homotopically nontrivial simple closed curve that is disjoint from its image. We prove that every rational point in the rotation set is realized by a periodic point provided that there is no free curve and the rotation set has empty interior. This gives a topological version of a theorem of Franks. Using this result, and inspired by a theorem of Guillou, we prove a version of the Poincaré-Birkhoff Theorem for torus homeomorphisms: in the abse…

Deviations of Ergodic sums for Toral Translations I : Convex bodies

We show the existence of a limiting distribution $\cD_\cC$ of the adequately normalized discrepancy function of a random translation on a torus relative to a strictly convex set $\cC$. Using a correspondence between the small divisors in the Fourier series of the discrepancy function and lattices with short vectors, and mixing of diagonal flows on the space of lattices, we identify $\cD_\cC$ with the distribution of the level sets of a function defined on the product of the space of lattices with an infinite dimensional torus. We apply our results to counting lattice points in slanted cylinder…

A proof of the invariant torus theorem of Kolmogorov

Kolmogorov's invariant torus theorem is proved using a simple fixed point theorem.

A connecting theorem for geodesic flows on the 2-torus

We use a result of J. Mather on the existence of connecting orbits for compositions of monotone twist maps of the cylinder to prove the existence of connecting geodesics on the unit tangent bundle $ST^2$ of the 2-torus in regions without invariant tori.

A generalization of the line translation theorem

Through the method of brick decomposition and the operations on essential topological lines, we generalize the line translation theorem of Beguin, Crovisier, Le Roux [BCL] in the case where the property of preserving a finite measure with total support is replaced by the intersection property.

Linear random walks on the torus

We prove a quantitative equidistribution result for linear random walks on the torus, similar to a theorem of Bourgain, Furman, Lindenstrauss and Mozes, but without any proximality assumption.

Central limit theorem and law of the iterated logarithm for the linear random walk on the torus

Let $ρ$ be a probability measure on $\mathrm{SL}\_d(\mathbb{Z})$ and consider the random walk defined by $ρ$ on the torus $\mathbb{T}^d = \mathbb{R}^d/\mathbb{Z}^d$. Bourgain, Furmann, Lindenstrauss and Mozes proved that under an assumption on the group generated by the support of $$ρ$$, the random walk starting at any irrational point equidistributes in the torus. In this article, we study the central limit theorem and the law of the iterated logarithm for this walk starting at some point having good diophantine properties.

The classical KAM theorem for Hamiltonian systems via rational approximations

In this paper, we give a new proof of the classical KAM theorem on the persistence of an invariant quasi-periodic torus, whose frequency vector satisfies the Bruno-Rüssmann condition, in real-analytic non-degenerate Hamiltonian systems close to integrable. The proof, which uses rational approximations instead of small divisors estimates, is an adaptation to the Hamiltonian setting of the method we introduced in a previous work for perturbations of constant vector fields on the torus.

Rank one and mixing differentiable flows

We construct, over some minimal translations of the two torus, special flows under a differentiable ceiling function that combine the properties of mixing and rank one.

Universal frequency-preserving KAM persistence via modulus of continuity

In this paper, we study the persistence and remaining regularity of KAM invariant torus under sufficiently small perturbations of a Hamiltonian function together with its derivatives, in sense of finite smoothness with modulus of continuity, as a generalization of classical Hölder continuous circumstances. To achieve this goal, we extend the Jackson approximation theorem to the case of modulus of continuity, and establish a corresponding regularity theorem adapting to the new iterative scheme. Via these tools, we establish a KAM theorem with sharp differentiability hypotheses, which asserts th…

Uniform convergence rate for Birkhoff means of certain uniquely ergodic toral maps

We obtain estimates on the uniform convergence rate of the Birkhoff average of a continuous observable over torus translations and affine skew product toral transformations. The convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency defining the transformation. Furthermore, we show that for the one dimensional torus translation, these estimates are nearly optimal.

Local Rigidity of Diophantine translations in higher dimensional tori

We prove a theorem asserting that, given a Diophantine rotation $α$ in a torus $\T ^{d} \equiv \R ^{d} / \Z ^{d}$, any perturbation, small enough in the $C^{\infty}$ topology, that does not destroy all orbits with rotation vector $α$ is actually smoothly conjugate to the rigid rotation. The proof relies on a K.A.M. scheme (named after Kolmogorov-Arnol'd-Moser), where at each step the existence of an invariant measure with rotation vector $α$ assures that we can linearize the equations around the same rotation $α$. The proof of the convergence of the scheme is carried out in the $C^{\infty}$ ca…

环面上偏微分方程的 Birkhoff 正规形定理

A Birkhoff Normal Form Theorem for Partial Differential Equations on torus

本文证明了环面上哈密顿偏微分方程的抽象 Birkhoff 正规形定理，并将其应用于非线性波动方程和非线性 Schrödinger 方程，在更低的 Sobolev 正则性指标下获得了多项式及次指数长时间稳定性。

非可积三维流形中测地流的一致性

Uniformity of geodesic flow in non-integrable 3-manifolds

本文研究非可积有限多立方体平移三维流形上测地流的均匀分布问题，提出分裂方法处理含一个可积方向的笛卡尔积情形，并结合分裂覆盖论证将结果推广到三个方向均不可积且满足特殊条件的流形。

Distribution of periodic torus orbits and Duke's theorem for cubic fields

We study periodic torus orbits on spaces of lattices. Using the action of the group of adelic points of the underlying tori, we define a natural equivalence relation on these orbits, and show that the equivalence classes become uniformly distributed. This is a cubic analogue of Duke's theorem about the distribution of closed geodesics on the modular surface: suitably interpreted, the ideal classes of a cubic totally real field are equidistributed in the modular 5-fold SL_3(Z)\SL_3(R)/SO_3(R). In particular, this proves (a stronger form of) the folklore conjecture that the collection of maximal…

Statistics of Square-tiled Surfaces: Symmetry and Short Loops

Square-tiled surfaces are a class of translation surfaces that are of particular interest in geometry and dynamics because, as covers of the square torus, they share some of its simplicity and structure. In this paper, we study counting problems that result from focusing on properties of the square torus one by one. After drawing insights from experimental evidence, we consider the implications between these properties and their frequency in each stratum of translation surfaces.

环面上含时哈密顿量粒子的全局 Nekhoroshev 定理

A global Nekhoroshev theorem for particles on the torus with time dependent Hamiltonian

本文证明了 $R^d\times T^d$ 中含时哈密顿量的全局 Nekhoroshev 定理，证明了未微扰系统作用量随时间的增长界限，并将其应用于受含时电磁场影响的带电粒子动力学。

Limit Theorems for Translation Flows

The aim of this paper is to obtain an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus (Theorem 1) and to give a limit theorem for these flows (Theorem 2).

Limit theorems for toral translations

We discuss some classical and recent results and open problems on the statistical behavior of ergodic sums above toral translations, and their applications to Diophantine approximations and to ergodic properties of systems related to quasi-periodic dynamics such as skew products, cylindrical cascades and special flows.

Long hitting time, slow decay of correlations and arithmetical properties

Let $τ_r(x,x_0)$ be the time needed for a point $x$ to enter for the first time in a ball $B_r(x_0)$ centered in $x_0$, with small radius $r$. We construct a class of translations on the two torus having particular arithmetic properties (Liouville components with intertwined denominators of convergents) not satisfying a logarithm law, i.e. such that for generic $x,x_0$ \liminf_{r\to 0} \frac{\log τ_r(x,x_0)}{-\log r} = \infty. By considering a suitable reparametrization of the flow generated by a suspension of this translation, using a previous construction by Fayad, we show the existence of a…

A Nekhoroshev type theorem for the nonlinear Schrödinger equation on the d-dimensional torus.

We prove a Nekhoroshev type theorem for the nonlinear Schrödinger equation $$ iu_t=-Δu+V\star u+\partial_{\bar u}g(u,\bar u)\, \quad x\in \T^d, $$ where $V$ is a typical smooth potential and $g$ is analytic in both variables. More precisely we prove that if the initial datum is analytic in a strip of width $ρ>0$ with a bound on this strip equals to $\eps$ then, if $\eps$ is small enough, the solution of the nonlinear Schrödinger equation above remains analytic in a strip of width $ρ/2$ and bounded on this strip by $C\eps$ during very long time of order $ \eps^{-α|\ln \eps|^β}$ for some constan…

Quenched limit theorems for random U(1) extensions of expanding maps

The Lyapunov spectra of random U(1) extensions of expanding maps on the torus were investigated in our previous work [NW2015]. Using the result, we extend the recent spectral approach for quenched limit theorems for expanding maps [DFGV2018] and hyperbolic maps [DFGV2019] to our partially hyperbolic dynamics. Quenched central limit theorems, large deviations principles and local central limit theorems for random U(1) extensions of expanding maps on the torus are proved via corresponding theorems for abstract random dynamical systems.

Two remarks on the Shrinking Target Property

We show that a translation of vector V on the torus has the monotone shrinking target property if and only if the vector V is of constant type. Then, using reparametrizations of linear flows, we show that there exist area preserving real analytic maps of the three torus that are mixing of all orders and do not enjoy the monotone shrinking target property.

Zero Measure Spectrum for Multi-Frequency Schrödinger Operators

Building on works of Berthé--Steiner--Thuswaldner and Fogg--Nous we show that on the two-dimensional torus, Lebesgue almost every translation admits a natural coding such that the associated subshift satisfies the Boshernitzan criterion. As a consequence we show that for these torus translations, every quasi-periodic potential can be approximated uniformly by one for which the associated Schrödinger operator has Cantor spectrum of zero Lebesgue measure. We also describe a framework that can allow this to be extended to higher-dimensional tori.

Deviations of ergodic sums for toral translations II. Boxes

We study the Kronecker sequence $\{nα\}_{n\leq N}$ on the torus ${\mathbb T}^d$ when $α$ is uniformly distributed on ${\mathbb T}^d.$ We show that the discrepancy of the number of visits of this sequence to a random box, normalized by $\ln^d N$, converges as $N\to\infty$ to a Cauchy distribution. The key ingredient of the proof is a Poisson limit theorem for the Cartan action on the space of $d+1$ dimensional lattices.

Limit theorems for translation flows (published version)

The aim of this paper is to obtain an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus (Theorem 1) and to give a limit theorem for these flows (Theorem 2).

Decomposition of random walk measures on the one-dimensional torus

The main result of this paper is a decomposition theorem for a measure on the one-dimensional torus. Given a sufficiently large subset $S$ of the positive integers, an arbitrary measure on the torus is decomposed as the sum of two measures. The first one $μ_1$ has the property that the random walk with initial distribution $μ_1$ evolved by the action of $S$ equidistributes very fast. The second measure $μ_2$ in the decomposition is concentrated on very small neighborhoods of a small number of points.