David B. Ellis

The topological dynamics of semigroup actions

In these notes we explore the fine structure of recurrence for semigroup actions, using the algebraic structure of compactifications of the acting semigroup.

Characteristic classes of transversely homogeneous foliations

The foliations studied in this paper have transverse geometry modeled on a homogeneous space G/H with transition functions given by the left action of G. It is shown that the characteristic classes for such a foliation are determined by invariants of a certain flat bundle. This is used to prove that when G is semisimple, the characteristic classes are rigid under smooth deformations, extending work of Brooks, Goldman and Heitsch.

An energy inequality for higher order linear parabolic operators and its applications

A generalization of the classical energy inequality is obtained for evolution operators (d¡dt)I—H(t)A.2k—J(t), associated with higher order linear parabolic operators with variable coefficients. Here H(t) and J(t) are matrices of singular integral operators. The key to the result is an algebraic inequality involving matrices similar to the symbol of H(t) having their eigenvalues contained in a fixed compact subset of the open left-half complex plane. Then a sharp estimate on the norms of certain imbedding maps is obtained. These estimates along with the energy inequality is applied to the Cauchy problem for higher order linear parabolic operators restricted to slabs in Rn + 1.

The regionally proximal relation

Sufficient conditions for the regionally proximal relation Q(X) of a minimal flow to be an equivalence relation are obtained in terms of the group 9(X) of the flow and various groups which depend only on the acting group T.

An icer approach to the theory of minimal flows

We study minimal flows by studying a universal minimal flow, invariant closed equivalence relations (icers) on it, subgroups of its group of automorphisms, and the interplay among these objects. As examples of this approach we discuss and give short proofs of some standard results on distal flows. We end with a statement of the Furstenberg structure theorem from this point of view.