Minimal topological actions do not determine the measurable orbit equivalence class
نویسندگان
چکیده
We construct an amenable action ˆ of a non-amenable group on a discrete space. This action extends to a minimal topological action ẑ of on a Cantor set C . We show that ẑ is non-uniquely ergodic and furthermore there exist ergodic invariant measures 1 and 2 such that . ẑ ; C; 1/ and . ẑ ; C; 2/ are not orbit equivalent measurable equivalence relations. This also provides an instance of the failure of equivalence between the notions of “global” and “local” amenability for countable equivalence relations. Mathematics Subject Classification (2000). 37A20, 43A07, 28D15, 37A15.
منابع مشابه
A ug 2 00 6 Minimal topological actions do not determine the measurable orbit equivalence class
We construct a minimal topological action Φ̃ of a non-amenable group on a Cantor set C, which is non-uniquely ergodic and furthermore there exist ergodic invariant measures μ1 and μ2 such that (Φ̃, C, μ1) and (Φ̃, C, μ2) are not orbit equivalent measurable equivalence relations. AMS Subject Classifications: 37A20
متن کاملA Bratteli Diagram for Commuting Homeomorphisms of the Cantor Set
This paper presents a construction of a sequence of Kakutani-Rohlin Towers and a corresponding simple Bratteli Diagram, B, for a general minimal aperiodic Z d action on a Cantor Set, X, with d 2. This is applied to the description of the orbit structure and to the calculation of the ordered group of coinvariants. For each minimal continuous Z d action on X there is a minimal continuous Z action...
متن کاملFull Groups and Orbit Equivalence in Cantor Dynamics
In this note we consider dynamical systems (X, G) on a Cantor set X satisfying some mild technical conditions. The considered class includes, in particular, minimal and transitive aperiodic systems. We prove that two such systems (X1, G1) and (X2, G2) are orbit equivalent if and only if their full groups are isomorphic as abstract groups. This result is a topological version of the well-known D...
متن کاملOrbit equivalence rigidity and bounded cohomology
We establish new results and introduce new methods in the theory of measurable orbit equivalence, using bounded cohomology of group representations. Our rigidity statements hold for a wide (uncountable) class of groups arising from negative curvature geometry. Amongst our applications are (a) measurable Mostow-type rigidity theorems for products of negatively curved groups; (b) prime factorizat...
متن کاملWhen Is a Group Action Determined by It’s Orbit Structure?
We present a simple approach to questions of topological orbit equivalence for actions of countable groups on topological and smooth manifolds. For example, for any action of a countable group Γ on a topological manifold where the fixed sets for any element are contained in codimension two submanifolds, every orbit equivalence is equivariant. Even in the presence of larger fixed sets, for actio...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008