David Fried

Three-dimensional affine crystallographic groups☆

Those groups r which act properly discontinuously and aillnely on II?’ with compact fundamental domain are classified. First it is shown that such a group f contains a solvable subgroup of finite index, thus establishing a conjecture of Auslander in dimension three. Then unimodular simply transitive alTine actions on IR’ are classified; this leads to a classification of atTine crystallographic groups acting on IR3. A characterization of which abstract groups admit such an action is given; moreover it is proved that every isomorphism between virtually solvable atline crystallographic groups (respectively simply transitive afline groups) is induced by conjugation by a polynomial automorphism of the affrne space. A characterization is given of which closed 3-manifolds can be represented as quotients of IR’ by groups of afftne transformations: a closed 3-manifold M admits a complete atline structure if and only if M has a finite covering homeomorphic (or homotopy-equivaient) to a 2-torus bundle over the circle.

Transitive Anosov flows and pseudo-Anosov maps

Abstract A TRANSITIVE Anosov flow on a closed manifold M is one with the qualitative behavior of a geodesic flow on a surface of negative curvature, that is global hyperbelocity and dense periodic set. A psedo-Anosov map is a homeomorphism of closed surface that has finitely many prescribed prong singlarities and is smooth and hyperbolic elsewhere: we refer to the Orsay Thurston Seminar for details [2]. We will show that Birkhoffs surfaces of section[1] can be used to established a close connection between these systems then M has dimension 3. This extends the srgery techniques of [4,5] to produce all the transitive Anove flows in dimension 3.

Flow equivalence, hyperbolic systems and a new zeta function for flows

We analyze the dynamics of diffeomorphisms in terms of their suspension flows. For many Axion A diffeomorphisms we find simplest representatives in their flow equivalence class and so reduce flow equivalence to conjugacy. The zeta functions of maps in a flow equivalence class are correlated with a zeta function ζH for their suspended flow. This zeta function is defined for any flow with only finitely many closed orbits in each homology class, and is proven rational for Axiom A flows. The flow equivalence of Anosov diffeomorphisms is used to relate the spectrum of the induced map on first homology to the existence of fixed points. For Morse-Smale maps, we extend a result of Asimov on the geometric index.

Affine manifolds and solvable groups

Let M be a compact affine manifold. Thus Af has a distinguished atlas whose coordinate changes are locally in Aff(&), the group of affine automorphisms of Euclidean w-space E. Assume M is connected and without boundary. The universal covering M of M has an affine immersion D: M—+E which is unique up to composition with elements of Aff(E). Corresponding to D there is a homomorphism a: n —> Aff (2?), where n is the group of deck transformations of Af, such that/) is equivariant for a. Set a(7r) = T. LetZ: Aff(E)~-* GL(E) be the natural map.

Subshifts on surfaces

We determine when a polynomial is the reduced zeta function of a basic set of a Smale diffeomorphism of a compact surface.

Complex dynamics of Möbius semigroups

We study the dynamics of semigroups of Mobius transformations on the Riemann sphere, especially their Julia sets and attractors. This theory relates to the dynamics of rational functions, rational semigroups, and Mobius groups and we compare and contrast these theories. We particularly examine Caruso’s family of Mobius semigroups, based on a random dynamics variant of the Fibonacci sequence.