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We introduce notions of continuous orbit equivalence and its one-sided version for countable left Ore semigroup actions on compact spaces by surjective local homeomorphisms, and characterize them in terms of the corresponding transformation groupoids and their operator algebras. In particular, we show that two essentially free semigroup actions on totally disconnected compact spaces are continuously orbit equivalent if and only if there is a canonical abelian subalgebra preserving C*-isomorphism between the associated transformation groupoid C*-algebras. We also give some examples of orbit equivalence, consider the special case of semigroup actions by homeomorphisms and relate continuous orbit equivalence of semigroup actions to that of the associated group actions. [ABSTRACT FROM AUTHOR]

We will characterize topologically conjugate one-sided topological Markov shifts in terms of their subgroups of continuous full groups and subalgebras of Cuntz–Krieger algebras. [ABSTRACT FROM AUTHOR]

We prove that any two continuous minimal (topologically free) actions of the infinite dihedral group on an infinite compact Hausdorff space are continuously orbit equivalent only if they are conjugate. We also show the above fails if we replace the infinite dihedral group by certain other virtually cyclic groups, e.g., the direct product of the integer group with any non-abelian finite simple group. [ABSTRACT FROM AUTHOR]

We will study several subgroups of continuous full groups of one-sided topological Markov shifts from the view points of cohomology groups of full group actions on the shift spaces. We also study continuous orbit equivalence and strongly continuous orbit equivalence in terms of these subgroups of the continuous full groups and the cohomology groups. [ABSTRACT FROM AUTHOR]

By the work of Brodzki–Niblo–Nowak–Wright and Monod, topological amenability of a continuous group action can be characterized using uniformly finite homology groups or bounded cohomology groups associated to this action. We show that (certain variations of) these groups are invariants for topologically free actions under continuous orbit equivalence. [ABSTRACT FROM AUTHOR]

A Cantor action is a minimal equicontinuous action of a countably generated group $G$ on a Cantor space $X$. Such actions are also called generalized odometers in the literature. In this work, we introduce two new conjugacy invariants for Cantor actions, the stabilizer limit group and the centralizer limit group. An action is wild if the stabilizer limit group is an increasing sequence of stabilizer groups without bound and otherwise is said to be stable if this group chain is bounded. For Cantor actions by a finitely generated group $G$ , we prove that stable actions satisfy a rigidity principle and furthermore show that the wild property is an invariant of the continuous orbit equivalence class of the action. A Cantor action is said to be dynamically wild if it is wild and the centralizer limit group is a proper subgroup of the stabilizer limit group. This property is also a conjugacy invariant and we show that a Cantor action with a non-Hausdorff element must be dynamically wild. We then give examples of wild Cantor actions with non-Hausdorff elements, using recursive methods from geometric group theory to define actions on the boundaries of trees. [ABSTRACT FROM AUTHOR]

A λ-graph system L is a labelled Bratteli diagram with shift operation. It is a generalized notion of finite labelled graph and presents a subshift. We will study continuous orbit equivalence of one-sided subshifts and topological conjugacy of two-sided subshifts from the view points of groupoids and C ∗ -algebras constructed from λ-graph systems. [ABSTRACT FROM AUTHOR]

Given an inverse semigroup S endowed with a partial action on a topological space X , we construct a groupoid of germs S ⋉ X in a manner similar to Exel's groupoid of germs, and similarly a partial action of S on an algebra A induces a crossed product A ⋊ S. We then prove, in the setting of partial actions, that if X is locally compact Hausdorff and zero-dimensional, then the Steinberg algebra of the groupoid of germs S ⋉ X is isomorphic to the crossed product A R (X) ⋊ S , where A R (X) is the Steinberg algebra of X. We also prove that the converse holds, that is, that under natural hypotheses, crossed products of the form A R (X) ⋊ S are Steinberg algebras of appropriate groupoids of germs of the form S ⋉ X. We introduce a new notion of topologically principal partial actions, which correspond to topologically principal groupoids of germs, and study orbit equivalence for these actions in terms of isomorphisms of the corresponding groupoids of germs. This generalizes previous work of the second-named author as well as from others, which dealt mostly with global actions of semigroups or partial actions of groups. We finish the article by comparing our notion of orbit equivalence of actions and orbit equivalence of graphs. [ABSTRACT FROM AUTHOR]

Abstract We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite type. This generalises a result of Matsumoto and Matui from the irreducible to the general case. We also prove that a pair of one-sided shift spaces of finite type are continuously orbit equivalent if and only if their groupoids are isomorphic, and that the corresponding two-sided shifts are flow equivalent if and only if the groupoids are stably isomorphic. As applications we show that two finite directed graphs with no sinks and no sources are move equivalent if and only if the corresponding graph C ⁎ -algebras are stably isomorphic by a diagonal-preserving isomorphism (if and only if the corresponding Leavitt path algebras are stably isomorphic by a diagonal-preserving isomorphism), and that two topological Markov chains are flow equivalent if and only if there is a diagonal-preserving isomorphism between the stabilisations of the corresponding Cuntz–Krieger algebras (the latter generalises a result of Matsumoto and Matui about irreducible topological Markov chains with no isolated points to a result about general topological Markov chains). We also show that for general shift spaces, strongly continuous orbit equivalence implies two-sided conjugacy. [ABSTRACT FROM AUTHOR]

For self-similar graph actions, we show that isomorphic inverse semigroups associated to a self-similar graph action are a complete invariant for the continuous orbit equivalence of inverse semigroup actions on infinite path spaces.

Drimbe and Vaes proved an orbit equivalence superrigidity theorem for left-right wreath product actions in the measurable setting. We establish the counterpart result in the topological setting for continuous orbit equivalence. This gives us minimal, topologically free actions that are continuous orbit equivalence superrigid. One main ingredient for the proof is to show continuous cocycle superrigidity for certain generalized full shifts, extending our previous result with Chung. [ABSTRACT FROM AUTHOR]

For self-similar graph actions, we show that isomorphic inverse semigroups associated to a self-similar graph action are a complete invariant for the continuous orbit equivalence of inverse semigroup actions on infinite path spaces. [ABSTRACT FROM AUTHOR]