From Auslander, page 5: If \(\phi : X \to X\) is a continuous map of a compact space \(X\), it can be “converted” to a homeomorphism by an “inverse limit” construction.

Let \(\tilde{X}\) be the 2-sequences of \(X\) for which \(\phi(x_{i+1}) = x_{i}\) for all \(i \in \mathbb{Z}\). Then \(\tilde{\phi}:\tilde{X} \to \tilde{X}\) defined by \(\tilde{\phi}(x) = x'\) where \(x'_{1} = \phi(x_{1})\). Then \(\tilde{\phi}\) is a homeomorphism of \(\tilde{X}\).