  We consider transformations of deterministic and random signals governed by
simple dynamical mappings. It is shown that the resulting signal can be a
random process described in terms of fractal distributions and fractal domain
integrals. In typical cases a steady state satisfies a dilatation equation,
relating an unknown function $f(x)$ to $f(\kappa x)$ (for example,
${f(x)=g(x)f(\kappa x)}$). We discuss simple linear models as well as nonlinear
systems with chaotic behavior including dissipative circuits with delayed
feedback.
