Self-similarity symmetry and fractal distributions in iterative dynamics of dissipative mappings

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.