Self-similar energies on self-similar graphs and sets

Self-similarity is a frequently used argument in mathematics and physics. We will focus on the scaling of a self-similar Dirichlet energy (the quadratic form of a Laplacian) on a self-similar graph. This imposes a self-similar geometry on the graph which allows a nontrivial heat conduction. The corresponding existence and uniqueness results can be phrased in the framework of nonlinear Perron-Frobenius Theory. Applied to the graphical skeleton of self-similar (finitely ramified) fractal set this technique enables us to couple certain fractals with different scaling laws. The resulting range of admissible scalings on the interface models different types of transition. Its analytic meaning is a scale of Besov spaces appearing as the trace (on the interface) of the set of functions of finite energy.