Differential transcendence of walks on self similar walks

Yakob Kahane (UQAM)

Symmetrically self-similar graphs are an important type of fractal graph. Their Green functions satisfy order one iterative functional equations. We show when the branching number of a generating cell is two, either the graph is a star consisting of finitely many one-sided lines meeting at an origin vertex, in which case the Green function is algebraic, or the Green function is differentially transcendental over C(z). The proof strategy relies on a recent work of Di Vizio, Fernandes and Mishna. The result adds evidence to a conjecture of Krön and Teufl about the spectrum of this family of graphs.

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