Rystsov I. Application of algebraic automata theory to analysis of discrete dynamical systems.

The thesis is devoted to the research in modern algebraic theory of automata. The focus of attention is done on Cerny conjecture, which asserts that in a synchronizing automaton with n states there is a reset word that translates all its states into a single state and whose length is at most 〖(n-1)〗^2. To achieve progress in this problem, author developed the theory of generalized linear and affine automata. The tight linear bound of Cerny function obtained for commutative and triangular automata, the tight sub-quadratic bound obtained for automata with zero, as well as the quadratic upper bound for regular automata and automata with simple idempotents. The criterion of primitiveness obtained for a finite automaton and proved that a primitive weakly defective finite automaton is irreducible.