1 Department of Mathematics, Science and Technology, Aarhus University2 Department of Mathematics, Science and Technology, Aarhus University
We consider a construction of $C^*$-algebras from continuous piecewise monotone maps on the circle which generalizes the crossed product construction for homeomorphisms and more generally the construction of Renault, Deaconu and Anantharaman-Delaroche for local homeomorphisms. Assuming that the map is surjective and not locally injective we give necessary and sufficient conditions for the simplicity of the $C^*$-algebra and show that it is then a Kirchberg algebra. We provide tools for the calculation of the K-theory groups and turn them into an algorithmic method for Markov maps.
Ergodic Theory and Dynamical Systems, 2015, Vol 35, Issue 2, p. 546-584