Prikarpatskii Ya. A.
Invariant measures for discrete dynamical systems and ergodic properties of generalized Boole-type transformations
Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 44-57
Invariant ergodic measures for generalized Boole-type transformations are studied using an invariant quasimeasure generating function approach based on special solutions for the Frobenius - Perron operator. New two-dimensional Boole-type transformations are introduced, and their invariant measures and ergodicity properties are analyzed.
Hamiltonian geometric connection associated with adiabatically perturbed Hamiltonian systems and the existence of adiabatic invariants
Ukr. Mat. Zh. - 2008. - 60, № 3. - pp. 382–387
Differential-geometric properties of the Hamiltonian connections on symplectic manifolds for the adiabatically perturbed Hamiltonian system are studied. Namely, the associated Hamiltonian connection on the main foliation is constructed and its description is given in terms of covariant derivatives and the curvature form of the corresponding connection.
Ukr. Mat. Zh. - 2006. - 58, № 6. - pp. 787–803
We develop a symplectic method for the investigation of invariant submanifolds of nonautonomous Hamiltonian systems and ergodic measures on them. The so-called Mel’nikov-Samoilenko problem for the case of adiabatically perturbed completely integrable oscillator-type Hamiltonian systems is studied on the basis of a new construction of “ virtual” canonical transformations.
Symplectic method for the construction of ergodic measures on invariant submanifolds of nonautonomous hamiltonian systems: Lagrangian manifolds, their structure, and mather homologies
Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 675–691
We develop a new approach to the study of properties of ergodic measures for nonautonomous periodic Hamiltonian flows on symplectic manifolds, which are used in many problems of mechanics and mathematical physics. Using Mather’s results on homologies of invariant probability measures that minimize some Lagrangian functionals and the symplectic theory developed by Floer and others for the investigation of symplectic actions and transversal intersections of Lagrangian manifolds, we propose an analog of a Mather-type ?-function for the study of ergodic measures associated with nonautonomous Hamiltonian systems on weakly exact symplectic manifolds. Within the framework of the Gromov-Salamon-Zehnder elliptic methods in symplectic geometry, we establish some results on stable and unstable manifolds for hyperbolic invariant sets, which are used in the theory of adiabatic invariants of slowly perturbed integrable Hamiltonian systems.
Ukr. Mat. Zh. - 2003. - 55, № 12. - pp. 1704-1719
On the basis of generalized Lagrange identity for pairs of formally adjoint multidimensional differential operators and a special differential geometric structure associated with this identity, we propose a general scheme of the construction of corresponding transformation operators that are described by nontrivial topological characteristics. We construct explicitly the corresponding integro-differential symbols of transformation operators, which are used in the construction of Lax-integrable nonlinear two-dimensional evolutionary equations and their Darboux–Bäcklund-type transformations.
Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable hamiltonian systems. I
Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1379–1390
By using the Cartan differential-geometric theory of integral submanifolds (invariant tori) of completely Liouville-Arnol’d integrable Hamiltonian systems on the cotangent phase space, we consider an algebraic-analytic method for the investigation of the corresponding mapping of imbedding of an invariant torus into the phase space. This enables one to describe analytically the structure of quasiperiodic solutions of the Hamiltonian system under consideration.