We prove that a Hamiltonian $p\in C^\infty(T^*{\bf R}^n)$ is locally integrable near a non-degenerate critical point $ρ_0$ of the energy, provided that the fundamental matrix at $ρ_0$ has no purely imaginary eigenvalues. This is done by using Birkhoff normal forms, which turn out to be convergent in the $C^\infty$ sense. We also give versions of the Lewis-Sternberg normal form near a hyperbolic fixed point of a canonical transformation. Then we investigate the almost holomorphic case.
We show that any analytically integrable Hamiltonian system near an equilibrium point admits a convergent Birkhoff normalization. The proof is based on a new, geometric approach to the problem.
The Bertrand-Darboux integrability condition for a certain class of perturbed harmonic oscillators is studied from the viewpoint of the Birkhoff-Gustavson(BG)-normalization: By solving an inverse problem of the BG-normalization on computer algebra, it is shown that if the perturbed harmonic oscillators with a homogeneous-{\it cubic} polynomial potential and with a homogeneous-{\it quartic} polynomial potentials admit the same BG-normalization up to degree-4 then both oscillators satisfy the Bertrand-Darboux integrability condition.
The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely, integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of almost every ellipse that preserves integrability near the boundary, is itself an ellipse. We apply this result to study local spectral rigidity of ellipses using the connection between the wave trace of the Laplacian and the dynamics near the boundary and establish rigidity for almost all of them.
We consider a convex curve $γ$ lying on the Sphere or Hyperbolic plane. We study the problem of existence of polynomial in velocities integrals for Birkhoff billiard inside the domain bounded by $γ$. We extend the result by S. Bolotin (1992) and get new obstructions on polynomial integrability in terms of the dual curve $Γ$. We follow a method which was introduced by S. Tabachnikov for Outer billiards in the plane and was applied later on in our recent paper to Birkhoff billiards with the help of a new the so called Angular billiard.
Among other results we show that near the equilibrium point, the Hamiltonian of the sine-Gordon (SG) equation on the circle can be viewed as an element in the Poisson algebra of the modified Korteweg-de Vries (mKdV) equation and hence by well established properties of the latter equation admits Birkhoff coordinates. On the other hand we prove that there exists a large set of smooth initial data, away from the equilibrium point and lying on the ramification locus of a double cover, for which the initial value problem of the SG equation has no classical solution.
We consider dynamical systems on compact manifolds, which are local diffeomorphisms outside an exceptional set (a compact submanifold). We are interested in analyzing the relation between the integrability (with respect to Lebesgue measure) of the first hyperbolic time map and the existence of positive frequency of hyperbolic times. We show that some (strong) integrability of the first hyperbolic time map implies positive frequency of hyperbolic times. We also present an example of a map with positive frequency of hyperbolic times at Lebesgue almost every point but whose first hyperbolic time …
We consider dynamical systems on compact manifolds, which are local diffeomorphisms outside an exceptional set (a compact submanifold). We are interested in analyzing the relation between the integrability (with respect to Lebesgue measure) of the first hyperbolic time map and the existence of positive frequency of hyperbolic times. We show that some (strong) integrability of the first hyperbolic time map implies the existence of a positive Lebesgue measure subset of points with positive frequency of hyperbolic times. We also present an example of a map with positive frequency of hyperbolic ti…
We consider some perturbations of a family of pairwise commuting linear quantum Hamiltonians on the torus with possibly dense pure point spectra. We prove that the Rayleigh-Schr{ö}dinger perturbation series converge near each unperturbed eigenvalue under the form of a convergent quantum Birkhoff normal form. Moreover the family is jointly diagonalised by a common unitary operator explicitly constructed by a Newton type algorithm. This leads to the fact that the spectra of the family remain pure point. The results are uniform in the Planck constant near $\hbar= 0$. The unperturbed frequencies s…
We consider dynamical systems on compact manifolds, which are local diffeomorphisms outside an exceptional set (a compact submanifold). We are interested in analyzing the relation between the integrability (with respect to Lebesgue measure) of the first hyperbolic time map and the existence of positive frequency of hyperbolic times. We show that some (strong) integrability of the first hyperbolic time map implies positive frequency of hyperbolic times. We also present an example of a map with positive frequency of hyperbolic times at Lebesgue almost every point but whose first hyperbolic time …
We prove that the geodesic flow on the unit tangent bundle to every hyperbolic 2-orbifold that is a sphere with 3 or 4 singular points admits explicit genus one Birkhoff sections, and we determine the associated first return maps.
We prove maximal inequalities for concentric ball and spherical shell averages on a general Gromov hyperbolic group, in arbitrary probability preserving actions of the group. Under an additional condition, satisfied for example by all groups acting isometrically and properly discontinuously on CAT(-1) spaces, we prove a pointwise ergodic theorem with respect to a sequence of probability measures supported on concentric spherical shells.
It is proved that the motion of a charge particle on a hyperbolic oriented two-dimensional surface in a magnetic field given by the volume form of the hyperbolic metric is completely integrable on the energy levels E < 1/2 in terms of real-analytic integrals. However it was known that on the level E=1/2 every trajectory is transitive and the restrictions of this flow onto the levels E>1/2 are Anosov flows
We prove there is a class of maps $γ:\mathbb{T}^{2n}\rightarrow\mathbb{S}^1$ such that a conservative dynamically coherent partially hyperbolic skew-product on $\mathbb{T}^{2n}\times\mathbb{S}^1$ with fixed hyperbolic dynamics on the base and rotation by angle $γ$ acting on the fibers have integrable hyperbolic structure which also implies in particular that they are not contact diffeomorphisms. In dimension $3$, we prove the same result using a standard technique in Contact Geometry, namely, that of \emph{characteristic foliations}, which gives a simple proof of the result but with more tight…
We show some level-2 large deviation principles for rational maps satisfying a strong form of non-uniform hyperbolicity, called "Topological Collet-Eckmann". More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages. For this purpose we show that each H{ö}lder continuous potential admits a unique equilibrium state, and that the pressure function can be characterized in terms of iterated preimages, periodic points, and Birkhoff averages. Then we use a variant of a general result of Kifer.
We prove, for f a partially hyperbolic diffeomorphism with center dimension one, two results about the integrability of its central bundle. On one side, we show that if the non wandering set of f is the whole manifold, and the manifold is 3 dimensional, then the absence of periodic points implies the unique integrability of the central bundle. On the opposite side, we prove that any periodic point p of large period n has an f n invariant center manifold, everywhere tangent to the center bundle. We also obtain, as a consequence of the last result, that there is an open and dense subset of C 1 r…
We prove a multifractal formalism for Birkhoff averages of continuous functions in the case of some non-uniformly hyperbolic maps, which includes interval examples such as the Manneville--Pomeau map.
Livšic theorem asserts that, for Anosov diffeomorphisms/flows, a Lipschitz observable is a coboundary if all its Birkhoff sums on every periodic orbits are equal to zero. The transfer function is then Lipschitz. We prove a positive Livšic theorem which asserts that a Lipschitz observable is bounded from below by a coboundary if and only if all its Birkhoff sums on periodic orbits are non negative. The new result is that the coboundary can be chosen Lipschitz. The map is only assumed to be $C^1$ and hyperbolic, but not necessarily bijective nor transitive. We actually prove our main result in t…
This paper compares different pseudo-Anosov maps coming from different Birkhoff sections of a given flow. More precisely, given a hyperbolic surface and a collection of periodic geodesics on it, we study those Birkhoff sections for the geodesic flow on the unit bundle to the surface bounded by the collection. We show that there is a canonical identification of all those surfaces, and that the first-return maps induced by the flow can all be expressed as a composition of negative Dehn twists along the same family of curves: only the order depends on the choice of a particular surface.
Given integers $g,n \geq 0$ satisfying $2-2g-n < 0$, let $\mathcal{M}_{g,n}$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $g$ with $n$ cusps. We study the global behavior of the Mirzakhani function $B \colon \mathcal{M}_{g,n} \to \mathbf{R}_{\geq 0}$ which assigns to $X \in \mathcal{M}_{g,n}$ the Thurston measure of the set of measured geodesic laminations on $X$ of hyperbolic length $\leq 1$. We improve bounds of Mirzakhani describing the behavior of this function near the cusp of $\mathcal{M}_{g,n}$ and deduce that $B$ is square-integrable wi…
A Reeb vector field satisfies the Kupka-Smale condition when all its closed orbits are non-degenerate, and the stable and unstable manifolds of its hyperbolic closed orbits intersect transversely. We show that, on a closed 3-manifold, any Reeb vector field satisfying the Kupka-Smale condition admits a Birkhoff section. In particular, this implies that the Reeb vector field of a $C^\infty$-generic contact form on a closed 3-manifold admits a Birkhoff section, and that the geodesic vector field of a $C^\infty$-generic Riemannian metric on a closed surface admits a Birkhoff section.
We consider billiard ball motion in a convex domain on a constant curvature surface influenced by the constant magnetic field. We examine the existence of integral of motion which is polynomial in velocities. We prove that if such an integral exists then the boundary curve of the domain determines an algebraic curve in $\mathbf{C}^3$ which must be nonsingular. Using this fact we deduce that for any domain different from round disc for all but finitely many values of the magnitude of the magnetic field billiard motion does not have Polynomial in velocities integral of motion.
We establish quantitative results for the statistical be\-ha\-vi\-our of \emph{infinite systems}. We consider two kinds of infinite system: i) a conservative dynamical system $(f,X,μ)$ preserving a $σ$-finite measure $μ$ such that $μ(X)=\infty$; ii) the case where $μ$ is a probability measure but we consider the statistical behaviour of an observable $ϕ\colon X\to[0,\infty)$ which is non-integrable: $\int ϕ\, dμ=\infty$. In the first part of this work we study the behaviour of Birkhoff sums of systems of the kind ii). For certain weakly chaotic systems, we show that these sums can be strongly …
Let $f$ be a conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism $A$ on $\mathbb{T}^3$. We show that the stable and unstable bundles of $f$ are jointly integrable if and only if every periodic point of $f$ admits the same center Lyapunov exponent with $A$. In particular, $f$ is Anosov. Thus every conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism on $\mathbb{T}^3$, is ergodic. This proves the Ergodic Conjecture proposed by Hertz-Hertz-Ures on $\mathbb{T}^3$.
In this article, we consider linearly convex complex cones in complex Banach spaces and we define a new projective metric on these cones. Compared to the hyperbolic gauge of Rugh, it has the advantage of being explicit, and easier to estimate. We prove that this metric also satisfies a contraction principle like Birkhoff's theorem for the Hilbert metric. We are thus able to improve existing results on spectral gaps for complex matrices. Finally, we compare the contraction principles for the hyperbolic gauge and our metric on particular cones, including complexification of Birkhoff cones. It ap…
We consider partially hyperbolic attractors for non-singular endomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. We prove volume lemmas for both Lebesgue measure on the topological basin of the attractor and the SRB measure supported on the attractor.As a consequence under a mild assumption we prove exponential large deviation bounds for the convergence of Birkhoff averages associated to continuous observables with respect to the SRB measure.
In 1968, V.I. Oseledets formulated the question of convergence in the Birkhoff theorem and the multiplicative ergodic theorem for measurable cocycles over flows under the condition of integrability for each individual t. A.M. Stepin and the author established (2016) the convergence along subsets of time of density 1. In this note, we show that moreover the convergence is fulfilled modulo time subsets of finite measure.
In this article we prove estimates for the topological pressure of the set of points whose Birkhoff time averages are far from the space averages corresponding to the unique equilibrium state that has a weak Gibbs property. In particular, if $f$ has an expanding repeller and $ϕ$ is an Hölder continuous potential we prove that the topological pressure of the set of points whose accumulation values of Birkhoff averages belong to some interval $I\subset \mathbb R$ can be expressed in terms of the topological pressure of the whole system and the large deviations rate function. As a byproduct we de…
In this paper we investigate periodic FPU chains with an even number of particles. We show that near the equilibrium point, any such chain admits a \emph{resonant} Birkhoff normal form of order four which is \emph{completely integrable} - an important fact which helps explain the numerical experiments of Fermi, Pasta, and Ulam. We analyze the moment map of the integrable approximation of an even FPU chain. Unlike in the case of odd FPU chains these integrable systems (generically) exhibit hyperbolic dynamics. As an application we prove that any FPU chain with Dirichlet boundary conditions admi…
We establish that, for every hyperbolic orbifold of type (2, q, $\infty$) and for every orbifold of type (2, 3, 4g+2), the geodesic flow on the unit tangent bundle is left-handed. This implies that the link formed by every collection of periodic orbits (i) bounds a Birkhoff section for the geodesic flow, and (ii) is a fibered link. We also prove similar results for the torus with any flat metric. Besides, we observe that the natural extension of the conjecture to arbitrary hyperbolic surfaces (with non-trivial homology) is false.
We prove an inequality for Hölder continuous differential forms on compact manifolds in which the integral of the form over the boundary of a sufficiently small, smoothly immersed disk is bounded by a certain multiplicative convex combination of the volume of the disk and the area of its boundary. This inequality has natural applications in dynamical systems, where Hölder continuity is ubiquitous. We give two such applications. In the first one, we prove a criterion for the existence of global cross sections to Anosov flows in terms of their expansion-contraction rates. The second application …
We prove statistical limit laws for sequences of Birkhoff sums of the type $\sum_{j=0}^{n-1}v_n\circ T_n^j$ where $T_n$ is a family of nonuniformly hyperbolic transformations. The key ingredient is a new martingale-coboundary decomposition for nonuniformly hyperbolic transformations which is useful already in the case when the family $T_n$ is replaced by a fixed transformation $T$, and which is particularly effective in the case when $T_n$ varies with $n$. In addition to uniformly expanding/hyperbolic dynamical systems, our results include cases where the family $T_n$ consists of intermittent …
Consider a homeomorphism $f$ defined on a compact metric space $X$ and a continuous map $ϕ\colon X \to \mathbb{R}$. We provide an abstract criterion, called \emph{control at any scale with a long sparse tail} for a point $x\in X$ and the map $ϕ$, that guarantees that any weak$\ast$ limit measure $μ$ of the Birkhoff average of Dirac measures $\frac1n\sum_0^{n-1}δ(f^i(x))$ is such that $μ$-almost every point $y$ has a dense orbit in $X$ and the Birkhoff average of $ϕ$ along the orbit of $y$ is zero. As an illustration of the strength of this criterion, we prove that the diffeomorphisms with nonh…
Consider a nonuniformly hyperbolic map $ T $ modelled by a Young tower with tails of the form $ O(n^{-β}) $, $ β>2 $. We prove optimal moment bounds for Birkhoff sums $ \sum_{i=0}^{n-1}v\circ T^i $ and iterated sums $ \sum_{0\le i<j<n}v\circ T^i\, w\circ T^j $, where $ v,w:M\to \Bbb{R}$ are (dynamically) Hölder observables. Previously iterated moment bounds were only known for $ β>5$. Our method of proof is as follows; (i) prove that $ T $ satisfies an abstract functional correlation bound, (ii) use a weak dependence argument to show that the functional correlation bound implies moment estimat…
In the general context of computable metric spaces and computable measures we prove a kind of constructive Borel-Cantelli lemma: given a sequence (constructive in some way) of sets $A_{i}$ with effectively summable measures, there are computable points which are not contained in infinitely many $A_{i}$. As a consequence of this we obtain the existence of computable points which follow the \emph{typical statistical behavior} of a dynamical system (they satisfy the Birkhoff theorem) for a large class of systems, having computable invariant measure and a certain ``logarithmic'' speed of convergen…
On a compact manifold of any dimension $d\geq 3$, we show that joint non-integrability of the stable and unstable foliation of a hyperbolic attractor with one-dimensional expanding direction, for a vector field of class $C^2$, implies exponential mixing with respect to its physical measure. Consequently, the set of Axiom A vector fields which mix exponentially with respect to the physical measure of its non-trivial attractors contains a $C^1$-open and $C^2$-dense subset of the set of all Axiom A vector fields. Moreover, for codimension one $C^2$ Anosov flows in any dimension $d\geq 3$, if the …
As a final work to establish that frame flows for geometrically finite hyperbolic manifolds of arbitrary dimensions are exponentially mixing with respect to the Bowen-Margulis-Sullivan measure, this paper focuses on the case with cusps. To prove this, we utilize the countably infinite symbolic coding of the geodesic flow of Li-Pan and perform a frame flow version of Dolgopyat's method à la Sarkar-Winter and Tsujii-Zhang. This requires the local non-integrability condition and the non-concentration property but the challenge in the presence of cusps is that the latter holds only on a large prop…
We prove a $C^\infty$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces. This is a consequence of a $C^\infty$ closing lemma for Reeb flows on closed contact three-manifolds, which was recently proved as an application of spectral invariants in embedded contact homology. A key new ingredient of this paper is an analysis of an area-preserving map near its fixed point, which is based on some classical results in Hamiltonian dynamics: existence of KAM invariant circles for elliptic fixed points, and convergence of the Birkhoff normal form for hyperbolic fixed points.
