Graph Maps Such As Degree Of Continuity

Here we main concern about the equicontinuity of a Graph( i . e. , one ?dimensional connected compact branched manifold). Suppose C is a Graph with k loops andf: G 梸- G is a continuous map with at least a peri- odic point. We prove the following three conditions are equivalent: (1) f is equicontinuous; (2) One of the following End ( C) + 2k + 1 conditions holds: in) F(Jm.⁰(E^G+2L)!) = flJ1(G),m = l,2,挃,End(G) J=1 + 2k; 1) 0 = flf(G) is a ioop on Gandf I^:0? 0 is conjugate j=1 to a rational rotation; (3) One of the following End ( C) + 2k + 1 conditions holds: k m) Lfl.m/^o(E^G+2i)! is uniformly convergent, m = 1,2, ,End(G)+2k; n) There is a poistive integer number n, is uniformly convergent. Here End ( C) denote the number of end points of C , and F (f) de- note the fixed point set of f...