Continuous orbit equivalence rigidity

By Xin Li. Ergodic Theory and Dynamical Systems, 2015

78 citations recorded by the source corpus.

We take the first steps towards a better understanding of continuous orbit equivalence, i.e., topological orbit equivalence with continuous cocycles. First, we characterize continuous orbit equivalence in terms of isomorphisms of $C^{\ast }$ -crossed products preserving Cartan subalgebras. This is the topological analogue of the classical result by Singer and Feldman-Moore in the measurable setting. Second, we turn to continuous orbit equivalence rigidity, i.e., the question whether for certain classes of topological dynamical systems, continuous orbit equivalence implies conjugacy. We show that this is not always the case by constructing topological dynamical systems (actions of free abelian groups and also non-abelian free groups) that are continuously orbit equivalent but not conjugate. Furthermore, we prove positive rigidity results. For instance, for solvable duality groups, general topological Bernoulli actions and certain subshifts of full shifts over finite alphabets are rigid.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e58979f3a7a34a00422c96b78edc2d1b49ef742f

How researchers see this paper

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In Algebra and Number Theory

1. Aidan Sims
2. Toke Meier Carlsen
3. Efren Ruiz
4. Xin Li
5. M. Tomforde
6. J. Renault
7. Trent Yeend
8. Iain Raeburn

In Mathematical Physics

1. Aidan Sims
2. Efren Ruiz
3. Toke Meier Carlsen
4. M. Tomforde
5. Xin Li
6. J. Renault
7. Trent Yeend
8. Iain Raeburn

Citations in the corpus

• Reconstruction of groupoids and C*-rigidity of dynamical systems (2017, background): “We detail the consequences of our results for dynamical systems, demonstrating the breadth of their applicability: we develop C-algebraic characterisations of appropriate notions of stabiliser-preserving orbit equivalence and topological conjugacy for group actions whose essential stabilisers are to”
• Every classifiable simple C*-algebra has a Cartan subalgebra (2018, background): “rivial pairing between K-theory and traces in [13]. Recently, the notion of Cartan subalgebras in C*-algebras [17, 26] has attracted some attention, due to connections to continuous orbit equivalence [18, 19] and the UCT question [1, 2]. In particular the reformulation of the UCT question in [1, 2] ”
• Cartan subalgebras in C*-algebras. Existence and uniqueness (2017, background): “Apart from being of interest for C*-algebra theory itself, the notion of Cartan subalgebras also builds a bridge to topological dynamical systems, as it is closely related to the notion of continuous orbit equivalence [32].”
• Partial transformation groupoids attached to graphs and semigroups (2016, background): “eover, “(ii) ⇒(i)” holds in general (i.e., without the assumption of topological freeness). Proof. The proof is completely analogous to the one of [22, Theorem 1.2]. Therefore, we refer the reader to [22] for details, and only mention the key ideas. For “(i) ⇒(ii)”, observe that G⋉X →H⋉Y,(g,x) →(a(g”