Let T be an invertible, ergodic, measure-preserving transformation of a separable, nonatomic probability space (X, (B, ra), and let U be the induced unitary operator acting in L(X, (B, m). Let Ct(T) be the Banach algebra generated by the multiplication algebra and the nonnegative powers of U. It is shown that, if 5 is another such transformation, then 5 and T are conjugate if, and only if, a(5)...

Three Notions of Classification Consider your favorite class ofmathematical structures, be it groups, modules, measure-preserving transformations, C-algebras, Lie groups, smooth manifolds, or something completely different. With some probability, the classification problem for these objects, that is, the problem of determining the structures up to some relevant notion of isomorphism, is, or has...

A nonsingular transformation is said to be doubly ergodic if for all sets A and B of positive measure there exists an integer n > 0 such that (T n(A) \ A) > 0 and (T n(A) \ B) > 0. While double ergodicity is equivalent to weak mixing for nite measure-preserving transformations, we show that this is not the case for in nite measure preserving transformations. We show that all measure-preserving ...

Let L1 be the set of all mappings f : Zp → Zp of the space of all p-adic integers Zp into itself that satisfy Lipschitz condition with a constant 1. We prove that the mapping f ∈L1 is ergodic with respect to the normalized Haar measure on Zp if and only if f induces a single cycle permutation on each residue ring Z/pZ modulo p, for all k = 1,2,3, . . .. The multivariate case, as well as measure...

The helicity of a vector field is a measure of the average linking of pairs of integral curves of the field. Computed by a six-dimensional integral, it is widely useful in the physics of fluids. For a divergence-free field tangent to the boundary of a domain in 3-space, helicity is known to be invariant under volume-preserving diffeomorphisms of the domain that are homotopic to the identity. We...