I know that fractal structures have power-laws in various forms "hidden" in them. I am looking for the most simple fractal model that I can find that generates time series with, say, Pareto-distributed increments. Or any other heavier-than-Gaussian -tailed distribution.
By generating time series, I mean that the system can be described by some kind of observations, any type. Temperature, some integral measure, derivative of some property, etc. For example, consider a system of agents organized in a network model, and define a sort of interaction between them on a graph. The observed measure of biggest cluster, or sum of vertex weights, or number of vertices with weight bigger than some specific number, etc., iterative, could be a time series.
By fractal properties I mean property of self-similarity. For example, having structure (for an agent-based model that could be graph structure of connectivities) that is defined by the same rule on all scales. For the mentioned exmaple of graph structure, that would be structure among clusters that copies structure of nodes inside every single cluster.
I am trying to come up with the most simple system that has fractal properties, demonstrates non-normality of the whole while having normality or binary behavior at the lowest level, and has clear connection between the two. Any help would be much appreciated!