$C^*$-algebras associated with reversible extensions of logistic maps

The construction of reversible extensions of dynamical systems presented in a previous paper by the author and A. V. Lebedev is enhanced, so that it applies to arbitrary mappings (not necessarily with open range). It is based on calculating the maximal ideal space of $C^*$-algebras that extends endomorphisms to partial automorphisms via partial isometric representations, and involves a new set of ‘parameters’ (the role of parameters is played by chosen sets or ideals).
As model examples, we give a thorough description of reversible extensions of logistic maps and a classification of systems associated with compression of unitaries generating homeomorphisms of the circle.
Bibliography: 34 titles.