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Fractals deal with special sets that exhibit complicated patterns in every scale. Fractal sets usually have a Hausdorff dimension different from its topological dimension. Examples include Julia sets, the Sierpinski triangle, the Cantor set. Fractals naturally appear in dynamical system, such as iterations in the complex plane, or as strange attractors to continuous dynamical systems, (see Lorentz attractor).
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A functional equation coming from a distribution function
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$$
I am interested in solving these functional equations or working on any special properties of $F$ or $\varphi$ (e.g., fractals or singularity). Any comment helps. …