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).
[F1985] Falconer K.J. The Geometry of Fractal Sets. Cambridge university press, 1985.
[BP2017] Bishop C.J., Peres Y. Fractals in probability and analysis. Cambridge; New York: Cambridge university press, 2017.