Iterated graph systems and the combinatorial Loewner property

Abstract

We introduce iterated graph systems which yield fractal spaces through a projective sequence of self-similar graphs. Our construction yields new examples of self-similar fractals, such as the pentagonal Sierpi\'nski carpet and a pillow-space, and it yields a new framework to study potential theory on many standard examples in the literature, including generalized Sierpi\'nski carpets. We demonstrate that a fractal arising from an iterated graph system satisfying mild geometric conditions and having enough reflection symmetries, satisfies the combinatorial Loewner property (CLP) of Bourdon and Kleiner. For certain exponents, one can also obtain the conductive homogeneity condition of Kigami. This shows, in a precise sense, that self-similarity and sufficient symmetry yields CLP. Furthermore, we show that our examples satisfy important asymptotic behaviors of discrete moduli. In particular, we establish the so-called super-multiplicative inequality for our examples - which when $p=2$ plays a crucial role in the study of Dirichlet forms and random walks on fractals. This implies the super-multiplicativity bound also for many fractals for which it was not known before, such as the Menger sponge. A particular feature here is using replacement flows and the notion of a flow basis. This yields a simpler way to establish many known estimates, and allows us to prove several new ones. Further, super-multiplicativity shows that one can effectively estimate the conformal dimensions numerically for many self-similar fractals.