Vladimir Ten

We propose and study a model for local dynamics of a perturbed convex real-analytic Liouville-integrable Hamiltonian system near a resonance of multiplicity $1+m$, $m \geqslant 0$. Physically, the model represents a toroidal pendulum, coupled with a Liouville-integrable system of $n$ non-linear rotators via a small analytic potential. The global bifurcation problem is set-up for the $n+1$ dimensional isotropic manifold, corresponding to a specific homoclinic orbit of the toroidal pendulum. The splitting of this manifold can be described by a scalar function on the $n$-torus. A sharp estimate for its Fourier coefficients is proven. It generalizes to a multiple resonance normal form of a convex analytic Liouville near-integrable Hamiltonian system. The bound then is exponentially small.

General theory for the splitting of separatrices near simple resonances of near-Liouville-integrable Hamiltonian systems is developed in the convex real-analytic setting. A generic estimate $$|\mathfrak{S}_k|\,\leqslant\,O(\sqrt{\varepsilon}) \, \times \, \exp\left[ -\,{\left| k\cdot\left(c_1{\omega\over\sqrt{\varepsilon}}+c_2\right)\right|} - |k|\sigma\right],\;\,k\in\mathbb{Z}^n\setminus\{0\}$$ is proved for the Fourier coefficients of the splitting distance measure $\mathfrak{S}(\phi),\,\phi\in\mathbb{T}^n,$ describing the intersections of Lagrangian manifolds, asymptotic to invariant $n$-tori, $\varepsilon$ being the perturbation parameter.
The constants $\omega\in\mathbb{R}^n,$ $c_1,\sigma>0,\,c_2\in\mathbb{R}^n$ are characteristic of the given problem (the Hamiltonian and the resonance), cannot be improved and can be calculated explicitly, given an example. The theory allows for optimal parameter dependencies in the smallness condition for $\varepsilon$.

New foundations of some aspects of statistical mechanics proposed.

We study polynomial in momenta integrals of geodesic flows on $D^2$. Some proclaims concerning orders of the integrals are proved.