Combinatorial and topological aspects of measure-preserving functions

We study measure-preserving functions between Lebesgue measurable subsets of the real line. We use particular bijections of the interval [0,1), called shifts, to approximate from below the set of measure-preserving maps on [0,1). This construction is similar to the method used in ergodic theory to obtain special transformations by cutting and stacking. In our approach we provide the set of shifts with an algebraic symmetric structure, which allows us to investigate the topic from both a combinatorial and a topological point of view. It is interesting the interplay between these two aspects of the problem.