Related papers: The Herman invariant tori conjecture

The double Hopf bifurcation and the existence of quasi-periodic invariant tori in a delayed van der Pol's oscillator are considered by regarding the damped coefficient and the delay as bifurcation parameters. Applying the center manifold…

For an integrable Hamiltonian systems with $d$ degrees of freedom ($d\geq 2$), we consider quantitatively the existence and non-existence of the flow-invariant Lagrangian torus with given frequency under the perturbation beyond the scope of…

We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang's Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures…

It is well known that a generic small perturbation of a Liouville-integrable Hamiltonian system causes breakup of resonant and near-resonant invariant tori. A general approach to the simple resonance case in the convex real-analytic setting…

We prove two special cases of a conjecture of J. Fern\'andez de Bobadilla for hypersurfaces with $1$-dimensional critical loci. We do this via a new numerical invariant for such hypersurfaces, called the beta invariant, first defined and…

Let $\Sigma$ be a $\sigma$-algebra over $\Omega$, and let $M(\Sigma)$ denote the Banach space of complex measures. Consider a representation $T_t$ for $t\in\Bbb R$ acting on $M(\Sigma)$. We show that under certain, very weak hypotheses,…

Moser's invariant tori for a class of nonanalytic quasi integrable even hamiltonian systems are shown to be analytic in the perturbation parameter. We do so by exhibiting a summation rule for the divergent series (``Lindstedt series") that…

In the paper the well known Riemann Hypothesis is proven. The proof is based on uniform approximation of the zeta function discs of the critical strip placed to the right from the critical line.The basic moment is a use of a new mesure…

This is an invitation to the probabilistic approach for constructing K\"ahler-Einstein metrics on complex projective algebraic manifolds X. The metrics in question emerge in the large N-limit from a canonical way of sampling N points on X,…

We adapt the Kolmogorov's normalization algorithm (which is the key element of the original proof scheme of the KAM theorem) to the construction of a suitable normal form related to an invariant elliptic torus. As a byproduct, our procedure…

In this note, we discuss various aspects of invariant measures for nonlinear Hamiltonian PDEs. In particular, we show almost sure global existence for some Hamiltonian PDEs with initial data of the form: "smooth deterministic function + a…

We generalize the Weinstein-Moser theorem on the existence of nonlinear normal modes (i.e., periodic orbits) near an equilibrium in a Hamiltonian system to a theorem on the existence of relative periodic orbits near a relative equilibrium…

Using algebraic transformations and equivalent reformulations we derive a number of new results from some earlier ones (by the author) in more accepted terms closely related to well-known conjectures of Bondy and Jung including a number of…

It is widespread since the beginning of KAM Theory that, under "sufficiently small" perturbation, of size $\epsilon$, apart a set of measure $O(\sqrt{\epsilon})$, all the KAM Tori of a non-degenerate integrable Hamiltonian system persist up…

We show that part I of uniform Martin's conjecture follows from a local phenomenon, namely that if a non-constant Turing invariant function goes from the Turing degree $\boldsymbol x$ to the Turing degree $\boldsymbol y$, then $\boldsymbol…

We extend a theorem of Herman from the case of unicritical polynomials to the case of polynomials with two finite critical values. This theorem states that Siegel disks of such polynomials, under a diophantine condition (called Herman's…

We present another view dealing with the Arnold-Givental conjecture on a real symplectic manifold $(M, \omega, \tau)$ with nonempty and compact real part $L={\rm Fix}(\tau)$. For given $\Lambda\in (0, +\infty]$ and $m\in\N\cup\{0\}$ we show…

In this note we study critical points of a variation of the action functional of classical mechanics, where the Hamiltonian term is retarded. Following a more than hundert and fifty year old paper by Carl Neumann we as well introduce Taylor…

Under explicit diophantine conditions on $(\alpha,\beta)\in\RR^2$, we prove that the local two-point correlations of the sequence given by the values $(m-\alpha)^2+\break (n-\beta)^2$, with $(m,n)\in\ZZ^2$, are those of a Poisson process.…

We develop regularity theory for critical points of variational integrals defined on Hessian spaces of functions on open, bounded subdomains of $\mathbb{R}^n$, under compactly supported variations. The critical point solves a fourth order…