Reversible extensions of irreversible dynamical systems: the $C^*$-method

A construction of a reversible extension of irreversible dynamical systems is presented. It is based on calculating the maximal ideal spaces of the $C^*$-algebras generated by these systems and the corresponding reversible extensions of endomorphisms. Connections between the objects that arise and dynamical systems of Smale horseshoe and other types are revealed.
Bibliography: 20 titles.