The crossed product construction provides a bridge between operator algebras and dynamical systems. For instance, it is known that two ergodic non singular dynamical systems are orbit equivalent if and only if their associated von Neumann crossed products are isomorphic. In the topological setting, it is known that two minimal dynamical systems on Cantor spaces are strong orbit equivalent if and only if their associated C* crossed products are isomorphic. In the literature there are examples of dynamical systems which are not orbit equivalent yet their associated C* crossed products are either isomorphic or have isomorphic Elliott invariant. These examples, however, deal with spaces which are either disconnected or have dimension greater than one. By considering irrational time homeomorphisms of suspensions, this book provides examples of dynamical systems on 1-dimensional compact connected metric spaces with the required property. The analysis in the book suggests that conditions to extend the given result about Cantor minimal systems might need to take into account the topological entropy of the systems involved.

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