Some results about inverse limits with set-valued bonding functions

In the paper, we give new results about dimension of inverse limits with upper semicontinuous bonding functions. We also construct an upper semicontinuous function f:[0,1]→2[0,1]f:[0,1]→2[0,1] with the following properties: (1) the graph of f is an arc, (2) the graph of f   is surjective, (3) the inverse limit lim←{[0,1],f}n=1∞ contains a simple closed curve, and (4) the dimension of lim←{[0,1],f}n=1∞ is 1. This answers a question by W.T. Ingram.