We study global behavior of the following max‐type difference equationx_n+1= max {1/x_n,A_n/x_n−1},n= 0, 1, …, where is a sequence of positive real numbers with 0 ≤ inf A_n≤ sup A_n< 1. The special case when is a periodic sequence with periodkandA_n∈ (0, 1) for everyn≥ 0 has been completely investigated by Y. Chen. Here we extend his results to the general case.

LetGbe a graph, andf:G→Gbe a continuous map with periodic points. In this paper we show that the following five statements are equivalent.(1)fis equicontinuous.(2) There exists some positive integerNsuch thatf^Nis uniformly convergent.(3)fisS-equicontinuous for some positive integer sequenceS= {n₁<n₂< …}.(4) Ω(x, f) =ω(x, f)for everyx∈G.(5)is a periodic map.