Let G be an infinite connected graph G = (V, E) where V is a countably infinite set and E is a subset of V x V - diagonal, satisfying 1) and 2): ----- 1) Every edge e in E has 2 endpoints, and any 2 vertices v,w in V bound at most one edge. 2) If for any v in V, deg(v) denotes the number of edges v belongs to, then for all v we have 2 <= deg(v) < oo. ----- Let C denote all functions V(G) —> reals: C = {f : V(G) —> R} For any v in V let the star star(v) of v be the set of vertices w at the other end of all edges vx containing v: star(v) = {x in V | vx is in E} Define A : C —> C by letting A(f) at a vertex v be the average of f(x) on the star of v: A(f)(v) = (Sum_{x in star(v)} f(x)) / |star(v)|. Question: Which f in C approach a limit under iteration of A: lim A^n(f) = f^ n—>oo for some f^ in C ??? —Dan