Speciality:
01.01.03 (Mathematical physics)
Keywords: dynamical systems; trajectory theory of dinamical systems; quadratic stochastic operators and processes; topological entropy of quadratic operators; random quadratic operators; the random model of heredityin random environment; the Gibbs states; lattice models of statistical mechanics.
Subject:
It was proved that any two ergodic countably-countinous sequences of measurable partitions are lacunarily isomorphic, whence any two ergodic countably-continuous partitions are isomorphic. The translation-invariant extemal Gibbs states for Ising and Potts models on the Cayley tree were described and continuum extremal Gibbs states for these models were constructed. The conditions when disordered phase in the ferromagnetic Potts model on the Bethe lattice be extreme were determined. A exact solution of Ising model with competing ternary and binary interactions on Cayley tree was obtained. The quadratic stochastic processes both commutative and noncommutative were determined, the several ergodic theorems for these processes were proved and correlation between its and Markov processes was established. The constructions of quadratic stochastic operators by means of Gibbs states and Haar measures were proposed and ergodicity of quadratic operators in the second case was proved. The models of heredity described by random quadratic stochastic operators in the random environment were studied. The topological entropy of some class of quadratic stochastic operators were colculeted. The following conjecture was formulated: any Volterra quadratic stochastic operator has zero topological entropy.
Main publications:
Ganikhodjaev N. N. On stochastic processes generated by quadratic operators // Jour. of Theor. Prob., 1990, 3(1), 51–70.
Ganikhodjaev N. N., Rozikov U. A. On disordered phase in the ferromagnetic Potts model on the Bethe lattice // Osaka journal Math. 2000. Vol. 37(2), 373–383.
