Invariant densities for composed piecewise fractional linear maps

Abstract

The starting point for this paper was the following problem: If we know the invariant densities for two maps S and T can we say something about the invariant density of the map $$U = T \circ S$$ ? This problem seems not to be touched on within ergodic theory. In this note we look at the inverse problem. Let the invariant density for $$U = T \circ S$$ be known. What can we say about invariant densities for S and T? We discuss a simple model, namely the class of all piecewise fractional linear maps with two branches on the unit interval [0, 1].