This paper establishes an extreme $C^k$ reducibility theorem of quasi-periodic $SL(2, \mathbb {R})$ cocycles in the local perturbative region, revealing both the essence of Eliasson [Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Comm. Math. Phys. 146 (1992), 447–482], and Hou and You [Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems. Invent. Math. 190 (2012), 209–260] in respectively the non-resonant and resonant cases. By paralleling further the reducibility process with the almost reducibility, we are able to acquire the least initial regularity as well as the least loss of regularity for the whole Kolmogorov–Arnold–Moser (KAM) iterations. This, in return, makes various spectral applications of quasi-periodic Schrödinger operators wide open.

It is conjectured that the only integrable metrics on the two-dimensional torus are Liouville metrics. In this paper, we study a deformative version of this conjecture: we consider integrable deformations of a non-flat Liouville metric in a conformal class and show that for a fairly large class of such deformations, the deformed metric is again Liouville. The principal idea of the argument is that the preservation of rational invariant tori in the foliation of the phase space forces a linear combination on the Fourier coefficients of the deformation to vanish. Showing that the resulting linear system is non-degenerate will then yield the claim. Since our method of proof immediately carries over to higher dimensional tori, we obtain analogous statements in this more general case. To put our results in perspective, we review existing results about integrable metrics on the torus.

The problem of non-integrability of the circular restricted three-body problem is very classical and important in the theory of dynamical systems. It was partially solved by Poincaré in the nineteenth century: he showed that there exists no real-analytic first integral which depends analytically on the mass ratio of the second body to the total and is functionally independent of the Hamiltonian. When the mass of the second body becomes zero, the restricted three-body problem reduces to the two-body Kepler problem. We prove the non-integrability of the restricted three-body problem both in the planar and spatial cases for any non-zero mass of the second body. Our basic tool of the proofs is a technique developed here for determining whether perturbations of integrable systems which may be non-Hamiltonian are not meromorphically integrable near resonant periodic orbits such that the first integrals and commutative vector fields also depend meromorphically on the perturbation parameter. The technique is based on generalized versions due to Ayoul and Zung of the Morales–Ramis and Morales–Ramis–Simó theories. We emphasize that our results are not just applications of the theories.

In this paper, we study centrally symmetric Birkhoff billiard tables. We introduce a closed invariant set $\mathcal {M}_{\mathcal {B}}$ consisting of locally maximizing orbits of the billiard map lying inside the region $\mathcal {B}$ bounded by two invariant curves of $4$-periodic orbits. We give an effective bound from above on the measure of this invariant set in terms of the isoperimetric defect of the curve. The equality case occurs if and only if the curve is a circle.

We prove a contact non-squeezing phenomenon on homotopy spheres that are fillable by Liouville domains with large symplectic homology: there exists a smoothly embedded ball in such a sphere that cannot be made arbitrarily small by a contact isotopy. These homotopy spheres include examples that are diffeomorphic to standard spheres and whose contact structures are homotopic to standard contact structures. As the main tool, we construct a new version of symplectic homology, called selective symplectic homology, that is associated to a Liouville domain and an open subset of its boundary. The selective symplectic homology is obtained as the direct limit of Floer homology groups for Hamiltonians whose slopes tend to $+\infty$ on the open subset but remain close to $0$ and positive on the rest of the boundary.

This paper mainly concerns the KAM persistence of the mapping $\mathscr {F}:\mathbb {T}^{n}\times E\rightarrow \mathbb {T}^{n}\times \mathbb {R}^{n}$ with intersection property, where $E\subset \mathbb {R}^{n}$ is a connected closed bounded domain with interior points. By assuming that the frequency mapping satisfies certain topological degree condition and weak convexity condition, we prove some Moser-type results about the invariant torus of mapping $\mathscr {F}$ with frequency-preserving under small perturbations. To our knowledge, this is the first approach to Moser's theorem with frequency-preserving. Moreover, given perturbed mappings over $\mathbb {T}^n$, it is shown that such persistence still holds when the frequency mapping and perturbations are only continuous about parameter beyond Lipschitz or even Hölder type. We also touch the parameter without dimension limitation problem under such settings.

Reflection in a strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve C whose tangent lines are reflected by the billiard to lines tangent to C. The famous Birkhoff conjecture states that the only strictly convex billiards with a foliation by closed caustics near the boundary are ellipses. By Lazutkin’s theorem, there always exists a Cantor family of closed caustics approaching the boundary. In the present paper, we deal with an open billiard, whose boundary is a strictly convex embedded (non-closed) curve $\gamma $. We prove that there exists a domain U adjacent to $\gamma $ from the convex side and a $C^\infty $-smooth foliation of $U\cup \gamma $ whose leaves are $\gamma $ and (non-closed) caustics of the billiard. This generalizes a previous result by Melrose on the existence of a germ of foliation as above. We show that there exists a continuum of above foliations by caustics whose germs at each point in $\gamma $ are pairwise different. We prove a more general version of this statement for $\gamma $ being an (immersed) arc. It also applies to a billiard bounded by a closed strictly convex curve $\gamma $ and yields infinitely many ‘immersed’ foliations by immersed caustics. For the proof of the above results, we state and prove their analogue for a special class of area-preserving maps generalizing billiard reflections: the so-called $C^{\infty }$-lifted strongly billiard-like maps. We also prove a series of results on conjugacy of billiard maps near the boundary for open curves of the above type.

We show that robustly transitive endomorphisms of a closed manifold must have a non-trivial dominated splitting or be a local diffeomorphism. This allows to get some topological obstructions for the existence of robustly transitive endomorphisms. To obtain the result, we must understand the structure of the kernel of the differential and the recurrence to the critical set of the endomorphism after perturbation.

The two main results in this paper concern the regularity of the invariant foliation of a $C^0$ -integrable symplectic twist diffeomorphism of the two-dimensional annulus, namely that (i) the generating function of such a foliation is $C^1$ , and (ii) the foliation is Hölder with exponent $\tfrac 12$ . We also characterize foliations by graphs that are straightenable via a symplectic homeomorphism and prove that every symplectic homeomorphism that leaves invariant all the leaves of a straightenable foliation has Arnol’d–Liouville coordinates, in which the dynamics restricted to the leaves is conjugate to a rotation. We deduce that every Lipschitz integrable symplectic twist diffeomorphisms of the two-dimensional annulus has Arnol’d–Liouville coordinates and then provide examples of ‘strange’ Lipschitz foliations by smooth curves that cannot be straightened by a symplectic homeomorphism and cannot be invariant by a symplectic twist diffeomorphism.

For an ascending HNN-extension $G*_{\psi }$ of a finitely generated abelian group G, we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal {A}^{G*_{\psi }}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag–Solitar groups $\mathrm {BS}(1,N)$ , $N\ge 2$ , for which our results imply that a $\mathrm {BS}(1,N)$ -subshift of finite type which contains a configuration with period $a^{N^\ell }\!, \ell \ge 0$ , must contain a strongly periodic configuration with monochromatic $\mathbb {Z}$ -sections. Then we study proper n-colorings, $n\ge 3$ , of the (right) Cayley graph of $\mathrm {BS}(1,N)$ , estimating the entropy of the associated subshift together with its mixing properties. We prove that $\mathrm {BS}(1,N)$ admits a frozen n-coloring if and only if $n=3$ . We finally suggest generalizations of the latter results to n-colorings of ascending HNN-extensions of finitely generated abelian groups.

Consider an analytic Hamiltonian system near its analytic invariant torus $\mathcal T_0$ carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at $\mathcal T_0$ is convergent and has a particular form: it is an analytic function of its non-degenerate quadratic part. We prove that in this case there is an analytic canonical transformation—not just a formal power series—bringing the Hamiltonian into its Birkhoff normal form.

We use feral pseudoholomorphic curves and adiabatic degeneration to prove an extended version of the so-called ‘almost existence result’ for regular compact Hamiltonian energy surfaces. That is, that for a variety of symplectic manifolds equipped with a Hamiltonian, almost every (non-empty) compact energy level has a periodic orbit.

A Birkhoff billiard is a system describing the inertial motion of a point mass inside a strictly convex planar domain, with elastic reflections at the boundary. The study of the associated dynamics is profoundly intertwined with the geometric properties of the domain: while it is evident how the shape determines the dynamics, a more subtle and difficult question is the extent to which the knowledge of the dynamics allows one to reconstruct the shape of the domain. This translates into many intriguing inverse problems and unanswered rigidity questions, which have been the focus of very active research in recent decades. In this paper we describe some of these questions, along with their connection to other problems in analysis and geometry, with particular emphasis on recent results obtained by the authors and their collaborators.

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 use Lerman’s contact cut construction to find a sufficient condition for Hamiltonian diffeomorphisms of compact surfaces to embed into a closed $3$ -manifold as Poincaré return maps on a global surface of section for a Reeb flow. In particular, we show that the irrational pseudorotations of the $2$ -disc constructed by Fayad and Katok embed into the Reeb flow of a dynamically convex contact form on the $3$ -sphere.

The aim of this article is to apply a Floer theory to study symmetric periodic Reeb orbits. We define positive equivariant wrapped Floer homology using a (anti-)symplectic involution on a Liouville domain and investigate its algebraic properties. By a careful analysis of index iterations, we obtain a non-trivial lower bound on the minimal number of geometrically distinct symmetric periodic Reeb orbits on a certain class of real contact manifolds. This includes non-degenerate real dynamically convex star-shaped hypersurfaces in ${\mathbb {R}}^{2n}$ which are invariant under complex conjugation. As a result, we give a partial answer to the Seifert conjecture on brake orbits in the contact setting.

We consider frieze sequences corresponding to sequences of cluster mutations for affine D- and E-type quivers. We show that the cluster variables satisfy linear recurrences with periodic coefficients, which imply the constant coefficient relations found by Keller and Scherotzke. Viewing the frieze sequence as a discrete dynamical system, we reduce it to a symplectic map on a lower dimensional space and prove Liouville integrability of the latter.

In this paper we consider the following nonlinear quasi-periodic system:

$$\begin{eqnarray}{\dot{x}}=(A+\unicode[STIX]{x1D716}P(t,\unicode[STIX]{x1D716}))x+\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})+h(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$

$A$

$d\times d$

$\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})$

$\unicode[STIX]{x1D716}$

$h(x,t,\unicode[STIX]{x1D716})=O(x^{2})$

$x\rightarrow 0$

$P,g$

$h$

$t$

$\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$

$\unicode[STIX]{x1D6FC}$

$\unicode[STIX]{x1D716}$

$$\begin{eqnarray}{\dot{x}}=(A+B_{\ast }(t))x+h_{\ast }(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$

$h_{\ast }(x,t,\unicode[STIX]{x1D716})=O(x^{2})~(x\rightarrow 0)$

$\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$

$\unicode[STIX]{x1D716}$

We prove that every non-degenerate Reeb flow on a closed contact manifold M admitting a strong symplectic filling W with vanishing first Chern class carries at least two geometrically distinct closed orbits provided that the positive equivariant symplectic homology of W satisfies a mild condition. Under further assumptions, we establish the existence of two geometrically distinct closed orbits on any contact finite quotient of M. Several examples of such contact manifolds are provided, like displaceable ones, unit cosphere bundles, prequantisation circle bundles, Brieskorn spheres and toric contact manifolds. We also show that this condition on the equivariant symplectic homology is preserved by boundary connected sums of Liouville domains. As a byproduct of one of our applications, we prove a sort of Lusternik–Fet theorem for Reeb flows on the unit cosphere bundle of not rationally aspherical manifolds satisfying suitable additional assumptions.

It is proved that a symplectic twist map of the cotangent bundle $T^{\ast }\mathbb{T}^{d}$ of the $d$-dimensional torus that is without conjugate points is $C^{0}$-integrable, that is  $T^{\ast }\mathbb{T}^{d}$ is foliated by a family of invariant $C^{0}$ Lagrangian graphs.