Some aspects of ergodic deformations of nonlinear Hamiltonian systems and locally homeomorphic metric spaces associated with them

Banakh Taras Onufriyovych; Prykarpatsky Anatoliy Karolyovych

doi:https://doi.org/10.31861/bmj2018.03.008

Banakh Taras Onufriyovych ¹ , Prykarpatsky Anatoliy Karolyovych ^2,3

¹ Department of Algebra, Topology and Fundamentals of Mathematics, Ivan Franko National University of Lviv, Lviv, 79000, Ukraine

² Department of Economic Cybernetics, Drohobych Ivan Franko state Pedagogical University, Drohobych, 82100, Ukraine

³ University of Mining and Metallurgy, Krakow, 30-059, Poland

DOI: https://doi.org/10.31861/bmj2018.03.008

Keywords: Hamiltonian systems, Lagrangian manifolds, homeomorphism, criterion of homeomorphism

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Abstract

The orbits of sslowly perturbed non-autonomous Hamiltonian systems and associated with them ergodic deformations of the Lagrangian manifolds are studied. Main results are based on teh J. Mather approach [18, 19] to constructing homologies of invariant probabilistic measures, mini- mizining some Lagrangian functionals, as well as on the Gromov-Salamon-Zehnder-Floer elliptic theory [7,9,12,20,26] of constructing invariant manifolds. There are built invariant submanifolds, which supporting invariant ergodic measures and having the metric space structure, allowing locally homeomorphic mappings. The problem of constructing effective criteria of their global homerphicity, posed by A.M. Samoilenko at studying ergodic deformations of nonlinear Hamiltonian systemns amd their adiabatic invariants, is analyzed. It is stated that a mapping $f:X→Y$ from a linearly connected Hausdorf space $X$ in one-connected (in particularly, contractable space) space $Y$ is a homomorphism if $f$ is a local homeomorphism and preimage $f^{-1}(y)$ of every point $y∈Y$ is a non empty compact set in $X$.

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ACS Style: Banakh, T.O.; Prykarpatsky, A.K. Some aspects of ergodic deformations of nonlinear Hamiltonian systems and locally homeomorphic metric spaces associated with them. Bukovinian Mathematical Journal. 2019, 6 https://doi.org/https://doi.org/10.31861/bmj2018.03.008
AMA Style: Banakh TO, Prykarpatsky AK. Some aspects of ergodic deformations of nonlinear Hamiltonian systems and locally homeomorphic metric spaces associated with them. Bukovinian Mathematical Journal. 2019; 6(3-4). https://doi.org/https://doi.org/10.31861/bmj2018.03.008
Chicago/Turabian Style: Taras Onufriyovych Banakh, Anatoliy Karolyovych Prykarpatsky. 2019. "Some aspects of ergodic deformations of nonlinear Hamiltonian systems and locally homeomorphic metric spaces associated with them". Bukovinian Mathematical Journal. 6 no. 3-4. https://doi.org/https://doi.org/10.31861/bmj2018.03.008

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