Simple closed curves in inverse limits with set-valued functions

The purpose of this paper is to show an example of an upper semicontinuous function \(f: [0,1]\rightarrow 2^{[0,1]}\) $f:[0,1]\to 2^{[0,1]}$ with the following properties: (1) The graph of f is an arc, (2) the inverse limit \(\underleftarrow{\lim }\;f\) $\underleftarrow{lim}f$ is a continuum, (3) the dimension of \(\underleftarrow{\lim }\;f\) $\underleftarrow{lim}f$ is 1, and (4) for each \(n\ge 3\) $n\ge 3$ , \(G' _{n}(f)\) $G_n^{\prime }(f)$ contains a simple closed curve and \(\underleftarrow{\lim }\;f\) $\underleftarrow{lim}f$ contains a simple closed curve.