Spacetime near the Planck scale as an aperiodic tiling

We propose a mapping of renormalization group flows to substitution tilings. We claim that this mapping allows to exhibit limit cycle solutions. We discuss the fractal structure of these limit cycles. The mapping is then applied in the context of two seemingly unrelated topics. The first is Efimov physics, for which we explicitly build the mapping and use it to show the fractal structure of its spectrum. The second is Quantum Einstein Gravity below the Planck scale, where we use the mapping to obtain an aperiodic tiling, thus indicating a fractal space-time structure. We finally emphasize an interesting relation between Quantum Einstein Gravity and Efimov physics.