Ergodicity of a Class of Cocycles Over Irrational Rotations

It is proved that if is irrational and L²(S¹) with o(l/n)then for each mZ\{0} the corresponding skew product is ergodic. The rigidity of specialflows over irrational rotations with roof functions whose Fouriercoefficients are in o(l/n) is also shown.     

Transformations conjugate to their inverses have even essential values

Let be an ergodic automorphism defined on a standard Borel probability space for which and are isomorphic. We study the structure of the conjugating automorphisms and attempt to gain information about the structure of . It was shown in Ergodic transformations conjugate to their inverses by involutions by Goodson et al. (Ergodic Theory and Dynamical Systems 16 (1996), 97--124) that if is ergodic having simple spectrum and isomorphic to its inverse, and if is a conjugation between and (i.e. satisfies ), then , the identity automorphism. We give a new proof of this result which shows even more, namely that for such a conjugation , the unitary operator induced by on must have a multiplicity function whose essential values on the ortho-complement of the subspace are always even. In particular, we see that can be weakly mixing, so the corresponding must have even maximal spectral multiplicity (regarding as an even number).

     

Ergodic properties of real cocycles and pseudo-homogeneous Banach spaces

Given an irrational rotation, in the space of real bounded variation functions it is proved that there are ergodic cocycles whose small perturbations remain ergodic; in fact, the set of ergodic cocycles has nonempty dense interior.

Given a pseudo-homogeneous Banach space and an irrational rotation, we study the set of elements satisfying the mean ergodic theorem. Once such a space is not homogeneous, we prove it is not reflexive and not separable. In ``natural" cases, up to -cohomology, the only elements satisfying the mean ergodic theorem are those from the closure of trigonometric polynomials.

For pseudo-homogeneous spaces admitting a Koksma's inequality ergodicity of the corresponding cylinder flows can be deduced from spectral properties of some circle extensions. In particular this is the case of Lebesgue spectrum (in the orthocomplement of the space of eigenfunctions) for the circle extension.

     

Piecewise Absolutely Continuous Cocycles Over Irrational Rotations

For an irrational rotation of the circle group T=R/Z and apiecewise absolutely continuous function f:TR, the unitary operatorVh(x)=e^2if(x)h(x+) on L²(T) is studied. It is shown that iff has a single discontinuity with non-integer jump then V is-weakly mixing for some with 0<||<1. In particular Vhas continuous singular spectrum. The property of -weak mixing(with possible change of the value of , 0<||<1) holdsfor all irrational rotations and, given , is stable under perturbationsof f by functions with sufficiently small O(1/n)-norm. On theother hand, there exists a piecewise linear function f withtwo non-integer jumps such that the spectrum of V is continuoussingular for one value of and Lebesgue for another.     

A note on the existence of a largest topological factor with zero entropy

Given a topological system on a -compact Hausdorff space and its factor we show the existence of a largest topological factor containing such that for each -invariant measure , . When a relative variational principle holds, .

     

On the disjointness problem for Gaussian automorphisms

If , are two Gaussian automorphisms, where and are concentrated on independent sets, then we have a dichotomy: either they are spectrally disjoint or they have a common factor. As an application, we construct non-rigid automorphisms which are spectrally determined.

     

Unbounded gaps for cocycles and invariant measures for their Mackey actions

We show that for a class of type -cocycles over a -action of type its Mackey action must change the type.

     

An algebraic property of joinings

We show that an ergodic automorphism is semisimple if and only if the set of ergodic self-joinings is a subsemigroup of the semigroup of self-joinings.