Daniel J. Rudolph

×2 and ×3 invariant measures and entropy

Let p and q be relatively prime natural numbers. Define T 0 and S 0 to be multiplication by p and q (mod 1) respectively, endomorphisms of [0,1). Let μ be a borel measure invariant for both T 0 and S 0 and ergodic for the semigroup they generate. We show that if μ is not Lebesgue measure, then with respect to μ both T 0 and S 0 have entropy zero. Equivalently, both T 0 and S 0 are μ-almost surely invertible.

Entropy and mixing for amenable group actions.

For i a countable amenable group consider those actions of i as measurepreserving transformations of a standard probability space, written asfT∞g∞2i acting on (X;F;„). We sayfT∞g∞2i has completely positive entropy (or simply cpe for short) if for any flnite and nontrivial partition P of X the entropy h(T;P) is not zero. Our goal is to demonstrate what is well known for actions of and even d , that actions of completely positive entropy have very strong mixing properties. Let Si be a list of flnite subsets of i. We say the Si spread if any particular ∞6 id belongs to at most flnitely many of the sets SiS i1 i . Theorem 0.1. For fT∞g∞2i an action of i of completely positive entropy and P any flnite partition, for any sequence of flnite sets Siµ i which spread we have

Absolutely continuous cocycles over irrational rotations

AbstractFor homeomorphisms

Ergodic behaviour of Sullivan's geometric measure on a geometrically finite hyperbolic manifold

\left( {z,w} \right)\mathop \to \limits^{T\varphi } \left( {z . e^{2xi\alpha } ,\varphi \left( z \right)w} \right)

Uniform endomorphisms which are isomorphic to a Bernoulli shift

(z, w ∈S1,α is irrational,ϕ:S1→S1) of the torusS1×S1 it is proved thatTϕ has countable Lebesgue spectrum in the orthocomplement of the eigenfunctions wheneverϕ is absolutely continuous with nonzero topological degree and the derivative ofϕ is of bounded variation. Some other cocycles with bounded variation are studied and generalizations of the above result to certain distal homeomorphisms on finite dimensional tori are presented.

Metrics and entropy for non-compact spaces

The purpose of this work is to study the joinings of simple systems. First the joinings of a simple system with another ergodic system are treated; then the pairwise independent joinings of three systems one of which is simple. The main results obtained are: (1) A weakly mixing simple system with no non-trivial factors with absolutely continuous spectral type is simple of all orders. (2) A weakly mixing system simple of order 3 is simple of all orders.

Asymptotically Brownian skew products give non-loosely BernoulliK-automorphisms

Sullivans geometric measure on a geometrically finite hyperbolic manifold is shown to satisfy a mean ergodic theorem on horospheres and through this that the geodesic flow is Bernoulli.

ℤ n and ℝ n cocycle extensions and complementary algebras

SummaryWe prove that a ℤd-action by automorphisms of a compact, abelian group is Bernoulli if and only if it has completely positive entropy. The key ingredients of the proof are the extension of certain notions of asymptotic block independence from ℤ-actions to ℤd-action and their equivalence with Bernoullicity, and a surprisingly close link between one of these asymptotic block independence properties for ℤd-actions by automorphisms of compact, abelian groups and the product formula for valuations on global fields.