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We study the problem of persistence of minimal invariant sets with smooth boundary for a certain class of discrete-time set-valued dynamical systems, naturally arising in the context of random dynamical systems with bounded noise. In particular, we introduce a single-valued map, the so-called boundary map, which has the property that a certain class of invariant submanifolds for this map is in one-to-one correspondence with invariant sets for the corresponding set-valued map. We show that minimal invariant sets with smooth boundary persist under small perturbations of the set-valued map, provided that the associated boundary map is normally hyperbolic at the unit normal bundle of the boundary.

We prove that analytic Hamiltonian dynamics on tori, annuli, or Euclidean space can be approximated by a composition of nonlinear shear maps where each of the shears depends only on the position or only on the momentum.

The motion of N particles interacting by a smooth repelling potential and confined to a compact d-dimensional region is proved to be, under mild conditions, non-ergodic for all sufficiently large energies. Specifically, choreographic solutions, for which all particles follow approximately the same path close to an elliptic periodic orbit of the single-particle system, are proved to be KAM stable in the high energy limit. Finally, it is proved that the motion of N repelling particles in a rectangular box is non-ergodic at high energies for a generic choice of interacting potential: there exists a KAM-stable periodic motion by which the particles move fast only in one direction, each on its own path, yet in synchrony with all the other parallel moving particles. Thus, we prove that for smooth interaction potentials the Boltzmann ergodic hypothesis fails for a finite number of particles even in the high energy limit at which the smooth system appears to be very close to the Boltzmann hard-sphere gas.

We consider $C^r$ $(r=3,\dots,\infty,\omega)$ diffeomorphisms with a generic homoclinic tangency to a hyperbolic periodic point, where this point has at least one complex (non-real) central multiplier and some explicit assumptions on central multipliers are satisfied so that the dynamics near the homoclinic tangency is not effectively one-dimensional. We prove that $C^1$-robust heterodimensional cycles of co-index one appear in any generic two-parameter $C^r$-unfolding of such a tangency. These heterodimensional cycles also have $C^1$-robust homoclinic tangencies.

We consider a -equivariant flow in with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit Γ. We provide criteria for the existence of stable and unstable invariant manifolds of Γ. We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schrödinger equations is considered.

The motion of N particles interacting by a smooth repelling potential and confined to a compact d-dimensional region is proved to be, under mild conditions, non-ergodic for all sufficiently large energies. Specifically, choreographic solutions, for which all particles follow approximately the same path close to an elliptic periodic orbit of the single-particle system, are proved to be KAM stable in the high energy limit. Finally, it is proved that the motion of N repelling particles in a rectangular box is non-ergodic at high energies for a generic choice of interacting potential: there exists a KAM-stable periodic motion by which the particles move fast only in one direction, each on its own path, yet in synchrony with all the other parallel moving particles. Thus, we prove that for smooth interaction potentials the Boltzmann ergodic hypothesis fails for a finite number of particles even in the high energy limit at which the smooth system appears to be very close to the Boltzmann hard-sphere gas.

We give an analytic proof of the existence of Shilnikov chaos in complex Ginzburg-Landau equation subject to a large third-order dispersion perturbation.

This supplement to the article Approximating multi-dimensional Hamiltonian flows by billiards contains the proofs of C$^{0}$ and C$^{r}$ - closeness theorems stated in the article.

This supplement to the article "Approximating multi-dimensional Hamiltonian flows by billiards" contains the proofs of C$^{0}$ and C$^{r}$ - closeness theorems stated in the article., Comment: 6 pages, 1 figure

We use an extended quantum field theory (EQFT) hep-th/9911168 to explore possible observational effects of the spacetime. It is shown that as it was expected the spacetime foam can provide quantum bose fields with a cutoff at very small scales, if the energy of zero - point fluctuations of fields is taken into account. It is also shown that EQFT changes the behaviour of massless fields at very large scales (in the classical region). We show that as $r\gg 1/\mu $ the Coulomb and Newton forces acquire the behaviour $\sim 1/r$ (instead of $1/r^{2}$)., Comment: Latex, 4 pages