A Combinatorial Characterization of the Critical Itineraries of a   Uniform Dynamical System
  For a function from the unit interval to itself with constant slope and one discontinuity, the itineraries of the point of discontinuity are called the critical itineraries. These critical itineraries play a significant role in the study of $\beta$-expansions (with positive or negative $\beta$) and fractal transformations. A combinatorial characterization of the critical itineraries of such functions is provided. 
Michael Barnsley, Wolfgang Steiner, and Andrew Vince
math.DS