Invariant densities for piecewise linear maps of the unit interval

We find an explicit formula for the invariant density h of an arbitrary eventually expanding piecewise linear map tau of an interval [0, 1]. We do not assume that the slopes of the branches are the same and we allow arbitrary number of shorter branches touching zero or touching one or hanging in between. The construction involves the matrix S which is defined in a way somewhat similar to the definition of the kneading matrix of a continuous piecewise monotonic map. Under some additional assumptions, we prove that if 1 is not an eigenvalue of S, then the dynamical system (tau, h . m) is ergodic with full support.