Linear expansions, strictly ergodic homogeneous cocycles and fractals

We consider a compact space Θ on whichR acts additively andR ₊ acts multiplicatively satisfying the distributive law. Moreover,R-action is strictly ergodic. Such Θ is constructed as a space of colored tilings corresponding to a weighted substitution, which is a kind of natural extension of thef-expansion for a piecewise linearf. We define a homogeneous cocycleF on Θ, which was called a cocycle with the scaling property in 2]. This is a realization of fractal functions which admit the continuous scalings. This also defines a self-similar process with strictly ergodic, stationary increments which has 0 entropy.