Abstract

In this paper we investigate the role of real-valued coboundaries for classifying of minimal homeomorphisms of the Cantor set. This work follows the work of Giordano, Putnam, and Skau who showed that one can use integer-valued coboundaries to characterize minimal homeomorphisms up to strong orbit equivalence. First, we prove a rigidity result. We show that there is an orbit equivalence between minimal Cantor systems which preserves real-valued coboundaries if and only if the systems are flip conjugate. Second, we investigate a real analogue of the dynamical unital ordered cohomology group studied by Giordano, Putnam and Skau. We show that, in general, isomorphism of our unital ordered vector space determines a weaker relation than strong orbit equivalence and we characterize this relation in a certain finite dimensional case. Finally, we consider isomorphisms of this vector space which preserve the cohomology subgroup. We show that such an isomorphism gives rise to a strictly stronger relation than strong orbit equivalence. In particular, it determines topological discrete spectrum, but does not determine systems up to flip conjugacy.

In this paper we investigate the role of real-valued coboundaries for classifying of minimal homeomorphisms of the Cantor set. This work follows the work of Giordano, Putnam, and Skau who showed that one can use integer-valued coboundaries to characterize minimal homeomorphisms up to strong orbit equivalence. First, we prove a rigidity result. We show that there is an orbit equivalence between minimal Cantor systems which preserves real-valued coboundaries if and only if the systems are flip conjugate. Second, we investigate a real analogue of the dynamical unital ordered cohomology group studied by Giordano, Putnam and Skau. We show that, in general, isomorphism of our unital ordered vector space determines a weaker relation than strong orbit equivalence and we characterize this relation in a certain finite dimensional case. Finally, we consider isomorphisms of this vector space which preserve the cohomology subgroup. We show that such an isomorphism gives rise to a strictly stronger relation than strong orbit equivalence. In particular, it determines topological discrete spectrum, but does not determine systems up to flip conjugacy.

In this paper we investigate the role of real-valued coboundaries for classifying of minimal homeomorphisms of the Cantor set. This work follows the work of Giordano, Putnam, and Skau who showed that one can use integer-valued coboundaries to characterize minimal homeomorphisms up to strong orbit equivalence. First, we prove a rigidity result. We show that there is an orbit equivalence between minimal Cantor systems which preserves real-valued coboundaries if and only if the systems are flip conjugate. Second, we investigate a real analogue of the dynamical unital ordered cohomology group studied by Giordano, Putnam and Skau. We show that, in general, isomorphism of our unital ordered vector space determines a weaker relation than strong orbit equivalence and we characterize this relation in a certain finite dimensional case. Finally, we consider isomorphisms of this vector space which preserve the cohomology subgroup. We show that such an isomorphism gives rise to a strictly stronger relation than strong orbit equivalence. In particular, it determines topological discrete spectrum, but does not determine systems up to flip conjugacy.

In this paper we investigate the role of real-valued coboundaries for classifying of minimal homeomorphisms of the Cantor set. This work follows the work of Giordano, Putnam, and Skau who showed that one can use integer-valued coboundaries to characterize minimal homeomorphisms up to strong orbit equivalence. First, we prove a rigidity result. We show that there is an orbit equivalence between minimal Cantor systems which preserves real-valued coboundaries if and only if the systems are flip conjugate. Second, we investigate a real analogue of the dynamical unital ordered cohomology group studied by Giordano, Putnam and Skau. We show that, in general, isomorphism of our unital ordered vector space determines a weaker relation than strong orbit equivalence and we characterize this relation in a certain finite dimensional case. Finally, we consider isomorphisms of this vector space which preserve the cohomology subgroup. We show that such an isomorphism gives rise to a strictly stronger relation than strong orbit equivalence. In particular, it determines topological discrete spectrum, but does not determine systems up to flip conjugacy.

In this paper we investigate the role of real-valued coboundaries for classifying of minimal homeomorphisms of the Cantor set. This work follows the work of Giordano, Putnam, and Skau who showed that one can use integer-valued coboundaries to characterize minimal homeomorphisms up to strong orbit equivalence. First, we prove a rigidity result. We show that there is an orbit equivalence between minimal Cantor systems which preserves real-valued coboundaries if and only if the systems are flip conjugate. Second, we investigate a real analogue of the dynamical unital ordered cohomology group studied by Giordano, Putnam and Skau. We show that, in general, isomorphism of our unital ordered vector space determines a weaker relation than strong orbit equivalence and we characterize this relation in a certain finite dimensional case. Finally, we consider isomorphisms of this vector space which preserve the cohomology subgroup. We show that such an isomorphism gives rise to a strictly stronger relation than strong orbit equivalence. In particular, it determines topological discrete spectrum, but does not determine systems up to flip conjugacy.

Authors