Applications of infinite dimensional and non-Archimedean dynamical systems to the theory of Gibbs measures (doctoral dissertation discussion)

The dissertation is devoted to study dynamics of linear and nonlinear operators over real and non-Archimedean vector spaces. Non-Archimedean analogues of some well-known theorems on hypercyclicity and supercyclicity of linear operators are proved. Some generalized weighted backward shift operators on non-Archimedean $c_0$ space is introduced. Criterions of hypercyclicity, supercyclicity and cyclicity of such kind of operators are obtained. The set of all translation-invariant generalized $p$-adic Gibbs measures for the $p$-adic Ising model on semi-infinite Cayley tree is described. Moreover, necessity and sufficiency conditions on boundedness of such kind of measures are obtained. The dynamics of $p$-adic (where $p \ge 3$) Potts–Bethe mapping is studied. It was shown that under some conditions on parameters the dynamics of $p$-adic Potts–Bethe mapping has a chaotic character. As a conclusion of that fact, we inferred the existence of any periodic Gibbs measure for the $p$-adic Potts model on semi-infinite Cayley tree. The surjectivity and orthogonal preserving property of the associated infinite dimensional nonlinear Markov operators are investigated. An example of an orthogonal preserving nonlinear Markov operator is provided, which is not necessary to be surjective (in finite dimensional setting these notions are equivalent), and therefore, we found some sufficient conditions for the operator to be surjective. Furthermore, an application of the obtained results has been given to the solvability of certain class of Hammerstein integral equations.