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We study the Kronecker sequence $\{n\alpha\}_{n\leq N}$ on the torus ${\mathbb T}^d$ when $\alpha$ 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., Comment: 56 pages. This is a revised and expanded version of the prior submissions

We prove that a $$C^k$$, $$k\ge 2$$ pseudo-rotation f of the disc with non-Brjuno rotation number is $$C^{k-1}$$-rigid. The proof is based on two ingredients: (1) we derive from Franks’ Lemma on free discs that a pseudo-rotation with small rotation number compared to its $$C^1$$ norm must be close to the identity map; (2) using Pesin theory, we obtain an effective finite information version of the Katok closing lemma for an area preserving surface diffeomorphism f, that provides a controlled gap in the possible growth of the derivatives of f between exponential and sub-exponential. Our result on rigidity, together with a KAM theorem by Russmann, allow to conclude that analytic pseudo-rotations of the disc or the sphere are never topologically mixing. Due to a structure theorem by Franks and Handel of zero entropy surface diffeomorphisms, it follows that an analytic conservative diffeomorphism of the disc or the sphere that is topologically mixing must have positive topological entropy.

International audience; We prove that steep real-analytic elliptic equilibrium points are exponentially stable, generalizing results which were known only under a convexity assumption. This proves the general case of a conjecture of Nekhoroshev. This result is also an important step in our proof that generically, both in a topological and measure-theoretical sense, equilibrium points are super-exponentially stable.

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., Comment: Paper published in Proc. Sympos. Pure Math., 89, Amer. Math. Soc., Providence, RI, 2015

Any \begin{document}$ C^d $\end{document} conservative map \begin{document}$ f $\end{document} of the \begin{document}$ d $\end{document}-dimensional unit ball \begin{document}$ {\mathbb B}^d $\end{document}, \begin{document}$ d\geq 2 $\end{document}, can be realized by renormalized iteration of a \begin{document}$ C^d $\end{document} perturbation of identity: there exists a conservative diffeomorphism of \begin{document}$ {\mathbb B}^d $\end{document}, arbitrarily close to identity in the \begin{document}$ C^d $\end{document} topology, that has a periodic disc on which the return dynamics after a \begin{document}$ C^d $\end{document} change of coordinates is exactly \begin{document}$ f $\end{document}.

We present examples of nearly integrable analytic Hamiltonian systems with several strong diffusion properties: topological weak mixing and diffusion at all times. These examples are obtained by AbC constructions with several frequencies.

We show that one-dimensional random walks in a quasi-periodic environment with Liouville frequency generically have an erratic statistical behavior. In the recurrent case we show that neither quenched nor annealed limit theorems hold and both drift and variance exhibit wild oscillations, being logarithmic at some times and almost linear at other times. In the transient case we show that the annealed Central Limit Theorem fails generically. These results are in stark contrast with the Diophantine case where the Central Limit Theorem with linear drift and variance was established by Sinai., Comment: 39 pages, 2 figures

A classical Borel–Cantelli Lemma gives conditions for deciding whether an infinite number of rare events will happen almost surely. In this article, we propose an extension of Borel–Cantelli Lemma to characterize the multiple occurrence of events on the same time scale. Our results imply multiple Logarithm Laws for recurrence and hitting times, as well as Poisson Limit Laws for systems which are exponentially mixing of all orders. The applications include geodesic flows on compact negatively curved manifolds, geodesic excursions on finite volume hyperbolic manifolds, Diophantine approximations and extreme value theory for dynamical systems.

International audience; 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 construct a $C^1$ symplectic twist map $g$ of the annulus that has an essential invariant curve $\Gamma$ such that $\Gamma$ is not differentiable and $g$ restricted to $\Gamma$ is minimal.

For area preserving C2 surface diffeomorphisms, we give an explicit finite information condition on the exponential growth of the number of Bowen's (n, δ)-balls needed to cover a positive proportion of the space, that is sufficient to guarantee positive topological entropy. This can be seen as an effective version of Katok's horseshoe theorem in the conservative setting. We also show that the analogous result is false in dimension larger than 3.

The following dichotomy for affine ℤ k \mathbb{Z}^{k} actions on the torus 𝕋 d {\mathbb{T}}^{d} , k , d ∈ ℕ k,d\in{\mathbb{N}} , is shown to hold: (i) The linear part of the action has no rank-one factors, and then the affine action is locally rigid. (ii) The linear part of the action has a rank-one factor, and then the affine action is locally rigid in a probabilistic sense if and only if the rank-one factors are trivial. Local rigidity in a probabilistic sense means that rigidity holds for a set of full measure of translation vectors in the rank-one factors.

We give the first example of a smooth volume preserving mixing dynamical system such that the discrete Schrödinger operators on the line defined with a potential generated by this system and a Hölder sampling function have almost surely a continuous spectrum.

When a Gevrey smooth perturbation is applied to a quasi-convex integrable Hamiltonian, it is known that the KAM invariant tori that survive are sticky, that is, doubly exponentially stable. We show by examples the optimality of this effective stability., Comment: 35 pages

Our aim in this note is to present some of the crucial contributions of Jean-Christophe Yoccoz to the theory of circle diffeomorphisms. We start with a short historical account before exposing Yoccoz' work. Then we give a brief description of the main conceptual and technical tools of the theory, with a focus on describing Yoccoz' work and contributions.

We construct on $\R^{2d}$, for any $d \geq 3$, smooth Hamiltonians having an elliptic equilibrium with an arbitrary frequency, that is not accumulated by a positive measure set of invariant tori. For $d\geq 4$, the Hamiltonians we construct have not any invariant torus of dimension $d$. Our examples are obtained by a version of the successive conjugation scheme {\it \`a la} Anosov-Katok.

We construct an increasing sequence of natural numbers $(m_{n})_{n=1}^{+\infty }$ with the property that $(m_{n}{\it\theta}[1])_{n\geq 1}$ is dense in $\mathbb{T}$ for any ${\it\theta}\in \mathbb{R}\setminus \mathbb{Q}$, and a continuous measure on the circle ${\it\mu}$ such that $\lim _{n\rightarrow +\infty }\int _{\mathbb{T}}\Vert m_{n}{\it\theta}\Vert \,d{\it\mu}({\it\theta})=0$. Moreover, for every fixed $k\in \mathbb{N}$, the set $\{n\in \mathbb{N}:k\nmid m_{n}\}$ is infinite. This is a sufficient condition for the existence of a rigid, weakly mixing dynamical system whose rigidity time is not a rigidity time for any system with a discrete part in its spectrum.

We show the existence of a limiting distribution \({\mathcal{D}_\mathcal{C}}\) of the adequately normalized discrepancy function of a random translation on a torus relative to a strictly convex set \({\mathcal{C}}\) . 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 \({\mathcal{D}_\mathcal{C}}\) 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 cylinders and to time spent in a given ball by a random geodesic on the flat torus.

We study the spectral measures of conservative mixing flows on the 2-torus having one degenerate singularity. We show that, for a sufficiently strong singularity, the spectrum of these flows is typically Lebesgue with infinite multiplicity. For this, we use two main ingredients: 1) a proof of absolute continuity of the maximal spectral type for this class of non-uniformly stretching flows that have an irregular decay of correlations, 2) a geometric criterion that yields infinite Lebesgue multiplicity of the spectrum and that is well adapted to rapidly mixing flows., 7 figures

International audience; We study the accumulation of an elliptic fixed point of a real analytic Hamiltonian by quasi-periodic invariant tori.We show that a fixed point with Diophantine frequency vector omega (0) is always accumulated by invariant complex analytic KAM-tori. Indeed, the following alternative holds: If the Birkhoff normal form of the Hamiltonian at the invariant point satisfies a Russmann transversality condition, the fixed point is accumulated by real analytic KAM-tori which cover positive Lebesgue measure in the phase space (in this part it suffices to assume that omega (0) has rationally independent coordinates). If the Birkhoff normal form is degenerate, there exists an analytic subvariety of complex dimension at least d + 1 passing through 0 that is foliated by complex analytic KAM-tori with frequency omega (0).This is an extension of previous results obtained in [1] to the case of an elliptic fixed point.

International audience; 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-analytic but more generally for Gevrey smooth systems.

We study the distribution modulo $1$ of the values taken on the integers of $r$ linear forms in $d$ variables with random coefficients. We obtain quenched and annealed central limit theorems for the number of simultaneous hits into shrinking targets of radii $n^{-\frac{r}{d}}$. By the Khintchine-Groshev theorem on Diophantine approximations, $\frac{r}{d}$ is the critical exponent for the infinite number of hits., Comment: 32 pages

We develop a new KAM scheme that applies to SL(2, \({{\mathbb R}}\)) cocycles with one frequency, irrespective of any Diophantine condition on the base dynamics. It gives a generalization of Dinaburg–Sinai’s theorem to arbitrary frequencies: under a closeness to constant assumption, the non-Abelian part of the classical reducibility problem can always be solved for a positive measure set of parameters.

We show that a finite number of commuting diffeomorphisms with simultane ously Diophantine rotation numbers are smoothly conjugated to rotations. This solves a problem raised by Moser.

An outer billard map F is defined outside a closed convex curve Γ in the following way. Let z be a point on the plane. Consider the supporting line L(z) from z to Γ such that Γ lies on the right of L. F (z) lies on L(z) so that the point of contact divides the segment [z, F (z)] in half. If Γ contains segments then F (z) is not defined if L(z) contains a segment. In this case F (z) is defined almost everywhere but it is discontinuous. In [14] Moser outlined the proof of the fact that if Γ is smooth and strictly convex then all trajectories are bounded (the complete proof (for C-curves) was later given in [3]). Moser also asked ([14, 15]) what happens if Γ is only piecewise smooth, mentioning in particular that even in the case where Γ is quadrangle, the question of boundedness of the orbits was open.

Nous presentons une preuve du dernier theoreme geometrique d'Herman qui affirme que, si un diffeomorphisme F de l'anneau possede la propriete d'intersection, alors toute courbe C°° F-invariante, sur laquelle le nombre de rotation de F est diophantien, est accumulee par un ensemble de mesure positive de courbes invariantes C°° sur lesquelles F est C°° -conjuguee a une rotation. Ceci implique en particulier la stabilite des points fixes elliptiques diophantiens des diffeomorphismes du plan qui preservent l'aire. Le caractere remarquable de ce theoreme est qu'il ne requiert aucune condition de torsion.

In this paper we introduce a new technique that allows us to investigate reducibility properties of smooth SL(2, R)-cocycles over irrational rotations of the circle beyond the usual Diophantine conditions on these rotations. For any given irrational angle on the base, we show that if the cocycle has bounded fibered products and if its fibered rotation number belongs to a set of full measure $\Sigma(\a)$, then the matrix map can be perturbed in the $C^\infty$ topology to yield a $C^\infty$-reducible cocycle. Moreover, the cocycle itself is almost rotations-reducible in the sense that it can be conjugated arbitrarily close to a cocycle of rotations. If the rotation on the circle is of super-Liouville type, the same results hold if instead of having bounded products we only assume that the cocycle is $L^2$-conjugate to a cocycle of rotations. When the base rotation is Diophantine, we show that if the cocycle is $L^2$-conjugate to a cocycle of rotations and if its fibered rotation number belongs to a set of full measure, then it is $C^\infty$-reducible. This extends a result proven in [5]. As an application, given any smooth SL(2, R)-cocycle over a irrational rotation of the circle, we show that it is possible to perturb the matrix map in the $C^\infty$ topology in such a way that the upper Lyapunov exponent becomes strictly positive. The latter result is generalized, based on different techniques, by Avila in [1] to quasiperiodic SL(2, R)-cocycles over higher-dimensional tori. Also, in the course of the paper we give a quantitative version of a theorem by L. H. Eliasson, a proof of which is given in the Appendix. This motivates the introduction of a quite general KAM scheme allowing to treat bigger losses of derivatives for which we prove convergence.

In this paper we investigate the exponential growth of products of two matrices . We prove, assuming A is a fixed hyperbolic matrix, that for Lebesgue almost every B, products of length n involving less than nα, 0 ≤ α 1.

We study a class of completely integrable Hamiltonian system with two degrees of freedom for which the perturbed flow displays, on some energy level, invariant sets that are laminations over Aubry-Mather sets of a Poincare section of the flow. Each one of these laminations carries a unique invariant probability measure for the flow and it is interesting therefore to understand the statistical properties of this measure. From a result of Kocergin in [13], we know that mixing is a priori impossible. In this paper, we investigate on the possible occurrence of weak mixing. The answer will essentially depend on the number of orbits of gaps in the Aubry-Mather set. More precisely, if the Aubry-Mather set has exactly one orbit of gaps and is hyperbolic then the special flow over it with any smooth ceiling function will be conjugate to a suspension with a constant ceiling function, failing hence to be weak mixing or even topologically weak mixing. To the contrary, if the Aubry-Mather set has more than one orbit of gaps with at least two in a general position then the special flow over it will in general be weak mixing.

We augment the $C^\infty $ conjugation approximation method with explicit estimates on the conjugacy map. This allows us to construct ergodic volume-preserving diffeomorphisms measure-theoretically isomorphic to any a priori given Liouville rotation on a variety of manifolds. In the special case of tori the maps can be made uniquely ergodic.

We provide sufficient conditions on a positive function so that the associated special flow over any irrational rotation is either weak mixing or $L^2$-conjugate to a suspension flow. This gives the first such complete classification within the class of Liouville dynamics. This rigidity coexists with a plethora of pathological behaviors.

We study the accumulation of an invariant quasi-periodic torus of a Hamiltonian flow by other quasi-periodic invariant tori. ¶ We show that an analytic invariant torus ${\mathcal{T}}_{0}$ with Diophantine frequency $\omega _{0}$ is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at ${\mathcal{T}}_{0}$ satisfies a Rüssmann transversality condition, the torus ${\mathcal{T}}_{0}$ is accumulated by Kolmogorov–Arnold–Moser (KAM) tori of positive total measure. If the Birkhoff normal form is degenerate, there exists a subvariety of dimension at least $d+1$ that is foliated by analytic invariant tori with frequency $\omega _{0}$ . ¶ For frequency vectors $\omega _{0}$ having a finite uniform Diophantine exponent (this includes a residual set of Liouville vectors), we show that if the Hamiltonian $H$ satisfies a Kolmogorov nondegeneracy condition at ${\mathcal{T}}_{0}$ , then ${\mathcal{T}}_{0}$ is accumulated by KAM tori of positive total measure. ¶ In four degrees of freedom or more, we construct for any $\omega _{0}\in{\mathbb{R}}^{d}$ , $C^{\infty}$ (Gevrey) Hamiltonians $H$ with a smooth invariant torus ${\mathcal{T}}_{0}$ with frequency $\omega _{0}$ that is not accumulated by a positive measure of invariant tori.

A smooth diffeomorphism is said to be distributionally uniquely ergodic (DUE for short) when it is uniquely ergodic and its unique invariant probability measure is the only invariant distribution (up to multiplication by a constant). Ergodic translations on tori are classical examples of DUE diffeomorphisms. In this article we construct DUE diffeomorphisms supported on closed manifolds different from tori, providing some counterexamples to a conjecture proposed by Forni in [For08]., 20 pages

For $\theta \in [0,1]$, we consider the map $T_\a: \T^2 \to \T^2$ given by $T_\theta(x,y)=(x+\theta,y+2x+\theta)$. The skew product $f_\a: \T^2 \times \C \to \T^2 \times \C$ given by $f_\theta(x,y,z)=(T_\theta(x,y),z+e^{2 \pi i y})$ generates the so called Weyl sums cocycle $a_\a(x,n) = \sum_{k=0}^{n-1} e^{2\pi i(k^2\theta+kx)}$ since the $n^{{\rm th}}$ iterate of $f_\a$ writes as $f_\a^n(x,y,z)=(T_\a^n(x,y),z+e^{2\pi iy} a_\a(2x,n))$. In this note, we improve the study developed by Forrest in \cite{forrest2,forrest} around the density for $x \in \T$ of the complex sequence ${\{a_\a(x,n)\}}_{n\in \N}$, by proving the ergodicity of $f_\theta$ for a class of numbers $\a$ that contains a residual set of positive Hausdorff dimension in $[0,1]$. The ergodicity of $f_\a$ implies the existence of a residual set of full Haar measure of $x \in \T$ for which the sequence ${\{a_\a(x,n) \}}_{n \in \N}$ is dense.

Arnold and Kochergin mixing conservative flows on surfaces stand as the main and almost only natural class of mixing transformations for which higher order mixing has not been established, nor disproved. Under suitable arithmetic conditions on their unique rotation vector, of full Lebesgue measure in the first case and of full Hausdorff dimension in the second, we show that these flows are mixing of any order. For this, we show that they display a generalization of the so called Ratner property on slow divergence of nearby orbits, that implies strong restrictions on their joinings, which in turn yields higher order mixing. This is the first case in which the Ratner property is used to prove multiple mixing outside its original context of horocycle flows and we expect our approach will have further applications.

International audience

We prove the existence of mixed spectrum C 1 reparame- terizations of any linear ow on T 2 with Liouville rotation number. For a restricted class of Liouville rotation numbers, we prove the existence of mixed spectrum real-analytic reparameterizations. 2000 Math. Subj. Class. 37A20, 37A45, 37C05.

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

We develop a new KAM scheme that applies to SL(2,R) cocycles with one frequency, irrespective of any Diophantine condition on the base dynamics. It gives a generalization of Dinaburg-Sinai's Theorem to arbitrary frequencies: under a closeness to constant assumption, the non-Abelian part of the classical reducibility problem can always be solved for a positive measure set of parameters., 16 pages

International audience

We give a geometric criterion that implies a singular maximal spectral type for a dynamical system on a Riemannian manifold. The criterion, which is based on the existence of fairly rich but localized periodic approximations, is compatible with mixing. Indeed, we check it for an ad hoc class of smooth mixing flows on $\mathbb T^3$ obtained from linear flows by time change and thus providing natural examples of mixing smooth diffeomorphisms and flows with purely singular spectra

We consider the linear cocycle $(T,A)$ induced by a measure preserving dynamical system $T:X \to X$ and a map $A:X \to \mathit{SL}(2,\mathbb{R})$. We address the dependence of the upper Lyapunov exponent of $(T,A)$ on the dynamics $T$ when the map $A$ is kept fixed. We introduce explicit conditions on the cocycle that allow to perturb the dynamics, in the weak and uniform topologies, to make the exponent drop arbitrarily close to zero. In the weak topology we deduce that if $X$ is a compact connected manifold, then for a $C^r$ ($r \ge 1$) open and dense set of maps $A$, either $(T,A)$ is uniformly hyperbolic for every $T$, or the Lyapunov exponents of $(T,A)$ vanish for the generic measurable $T$. For the continuous case, we obtain that if $X$ is of dimension greater than 2, then for a $C^r$ ($r \ge 1$) generic map $A$, there is a residual set of volume-preserving homeomorphisms $T$ for which either $(T,A)$ is uniformly hyperbolic or the Lyapunov exponents of $(T,A)$ vanish., Comment: To appear on the Bulletin of the Brazilian Mathematical Society