There is a long tradition for constructing C*-algebras from dynamical systems. An important motivation for doing this is to get new examples of C*-algebras that can be studied via dynamical systems, but sometimes it is also possible to recover a dynamical systems from its C*-algebra.

 

C*-rigidity of dynamical systems is the principal that dynamical systems can be recovered, up to a suitable notion of equivalence, from C*-algebraic data associated to them. An example of this is the result of Giordano, Putnam, and Skau that says that the crossed products of two Cantor minimal systems are isomorphic if and only if the Cantor minimal systems are strong orbit equivalent. Another example is the result by Tomiyama that says that the crossed products of two topologically transitive dynamical systems on compact metric spaces are isomorphic in a diagonal-preserving way if and only if the systems are flip conjugate. 

 

Recently, it has been shown that it is possible to recover shifts of finite type up to flow equivalence, continous orbit equivalence, and conjugacy from their Cuntz-Krieger algebras. 

 

I will give an overview of these results and explain how groupoids can be used to prove and generalise them.

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