Factor orbit equivalence of compact group extensions

LetG be a compact metrizable group. We show that any two ergodic extensions of transformationsT ₁andT ₂ by rotations ofG are factor orbit equivalent relative toT ₁andT ₂, and the equivalence may be taken to have a certain natural form.     

Topological realizations of families of ergodic automorphisms, multitowers and orbit equivalence

We study minimal topological realizations of families of ergodic measure preserving automorphisms (e.m.p.a.'s). Our main result is the following theorem. Theorem: Let {T_p:p∈I} be an arbitrary finite or countable collection of e.m.p.a.'s on nonatomic Lebesgue probability spaces (Y _p v _p ). Let S be a Cantor minimal system such that the cardinality of the set ε _S of all ergodic S-invariant Borel probability measures is at least the cardinality of I. Then for any collection {μ _p :pεI} of distinct measures from ε _S there is a Cantor minimal system S′ in the topological orbit equivalence class of S such that, as a measure preserving system, (S ¹,μ _p ) is isomorphic to T_p for every p∈I. Moreover, S′ can be chosen strongly orbit equivalent to S if and only if all finite topological factors of S are measure-theoretic factors of T_p for all p∈I. This result shows, in particular, that there are no restrictions at all for the topological realizations of countable families of e.m.p.a.'s in Cantor minimal systems. Namely, for any finite or countable collection {T ₁,T₂,…} of e.m.p.a.'s of nonatomic Lebesgue probability spaces, there is a Cantor minimal systemS, whose collection {μ1,μ2…} of ergodic Borel probability measures is in one-to-one correspondence with {T ₁,T₂,…}, and such that (S,μ _i ) is isomorphic toT _i for alli. Furthermore, since realizations are taking place within orbit equivalence classes of a given Cantor minimal system, our results generalize the strong orbit realization theorem and the orbit realization theorem of [18]. Those theorems are now special cases of our result where the collections {T _p}, {T _p }{μ _p } consist of just one element each. Research of I.K. was supported by NSF grant DMS 0140068.     

The structure of ergodic measures for compact group extensions

In this paper, the structure of the set of invariant measures on a transformation group which is a free compact abelian group extension of another transformation group is studied from both the geometric and analytic viewpoints. It is shown in general that genuine ergodic decompositions are obtained in the non-metric setting for measures that project onto an ergodic measure. In addition, when all the spaces involved are metric, there is a structure theorem for all ergodic measures in terms of the ergodic measures on the base and naturally defined subgroups.     

On orbit equivalence of borel automorphisms

LetE andF be two Borel sets of the countable productZ of the two point space {0,1}. Assume thatE andF are invariant sets for the odometer transformationR and thatE andF are of measure zero with respect to the unique finiteR-invariant measure onZ. We show thatE andF areR-orbit equivalent in a strict sense.     

On classification of linear cocycles over ergodic automorphisms

Complex linear cocycles over ergodic automorphisms are classified with the help of the barycenter method. A adjoining random matrix is built in an explicit form.     

The structure of skew products with ergodic group automorphisms

We prove that ergodic automorphisms of compact groups are Bernoulli shifts, and that skew products with such automorphisms are isomorphic to direct products. We give a simple geometric demonstration of Yuzvinskii’s basic result in the calculation of entropy for group automorphisms, and show that the set of possible values for entropy is one of two alternatives, depending on the answer to an open problem in algebraic number theory. We also classify those algebraic factors of a group automorphism that are complemented.     

Recurrence in unipotent groups and ergodic nonabelian group extensions

LetT be a measure-preserving and ergodic transformation of a standard probability space (X,S, μ) and letf:X → SUT _d (ℝ) be a Borel map into the group of unipotent upper triangulard ×d matrices. We modify an argument in [12] to obtain a sufficient condition for the recurrence of the random walk defined byf, in terms of the asymptotic behaviour of the distributions of the suitably scaled mapsf(n,x)=(fT ⁿ⁻¹·fT ⁿ⁻²…fT·f). We give examples of recurrent cocycles with values in the continuous Heisenberg group H₁(ℝ)=SUT₃(ℝ), and we use a recurrent cocycle to construct an ergodic skew-product extension of an irrational rotation by the discrete Heisenberg group H₁(ℤ)=SUT₃(ℤ). The author was partially supported by the FWF research project P16004-MAT.     

A simple proof of the bernoullicity of ergodic automorphisms on compact abelian groups

Recently it was proved by D. Lind, and G. Miles and K. Thomas that every ergodic automorphism of a compact metric abelian group is Bernoullian. They reduce the problem to the finite-dimensional compact connected abelian group (solenoidal group), and then they use difficult methods in proving the case. By using ideas of Y. Katznelson we can give a proof, which is much simpler than the other extant proofs, for the solenoidal case.     

On local automorphisms of group algebras of compact groups

We show that with few exceptions every local isometric automorphism of the group algebra of a compact metric group is an isometric automorphism.

     

Remainders in extensions and finite unions of locally compact sets

We discuss the relationship between properties of spaces and their remainders in extensions from the class P _fin of all finite unions of locally compact spaces. In particular, we show that a space X ∈ P _fin iff the remainder in each (some) compactification of X is in P _fin. Then we study the class P _fin and the relationship between the remainders of a space from this class in compact extensions and give a generalization of the theorem of Henriksen-Isbell.     

Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids

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.     

Normal ergodic actions and extensions

We demonstrate that normal ergodic extensions of group actions are characterized as skew product actions given by cocycles into locally compact groups. As a consequence, Robert Zimmer’s characterization of normal ergodic group actions is generalized to the noninvariant case. We also obtain the uniqueness theorem which generalizes the von Neumann Halmos uniqueness theorem and Zimmer’s uniqueness theorem for normal actions with relative discrete spectrum.     

Orbit equivalence rigidity for ergodic actions of the mapping class group

We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher complexity. We prove similar rigidity results for a finite direct product of mapping class groups as well.      

A class of compact group extensions of Gauss transformation

We investigate the dynamical properties of a class of compact group extensions of Gauss transformation.