6 M ay 2 00 3 Fräıssé Limits , Ramsey Theory , and Topological Dynamics of Automorphism Groups

(A) We study in this paper some connections between the Fra¨ıssé theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. A prime concern of topological dynamics is the study of continuous actions of (Hausdorff) topological groups G on (Hausdorff) compact spaces X. These are usually referred to as (compact) G-flows. Of particular interest is the study of minimal G-flows, those for which every orbit is dense. Every G-flow contains a minimal subflow. A general result of topological dynamics asserts that every topological group G has a universal minimal flow M(G), a minimal G-flow which can be homomorphically mapped onto any other minimal G-flow. Moreover, this is uniquely determined, by this property, up to isomorphism. (As usual a homomorphism π : X → Y between G-flows is a continuous G-map and an isomorphism is a bijective homomorphism.) For separable, metrizable groups G, which are the ones that we are interested in here, the universal minimal flow of G is an inverse limit of manageable, i.e., metrizable G-flows, but itself may be very complicated, for example non-metrizable. In fact, for the " simplest " infinite G, i.e., the countable discrete ones, M(G) is a very complicated compact G-invariant subset of the space βG of ultrafilters on G and is always non-metrizable. Rather remarkably, it turned out that there are topological groups G for which M(G) is actually trivial, i.e., a singleton. This is equivalent to saying that G has a very strong fixed point property, namely every G-flow has a fixed point (i.e., a point x such that g · x = x, ∀g ∈ G). (For separable, 1 metrizable groups this is also equivalent to the fixed point property restricted to metrizable G-flows.) Such groups are said to have the fixed point on com-pacta property or be extremely amenable. The latter name comes from one of the standard characterizations of second countable locally compact amenable groups. A second countable locally compact group G is amenable iff every metrizable G-flow has an invariant (Borel probability) measure. However, no locally compact group can be extremely amenable, because, by a theorem of Veech [77], every such group admits a free G-flow (i.e., a flow for which g · x = x ⇒ g = 1 G). This probably explains the rather late emergence of extreme amenability. The first examples of extremely amenable groups were constructed …