2 4 Ju n 20 04 FRAÏSSÉ 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 (Haus-dorff) 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 non-trivial 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, 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 compacta 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 non-trivial locally compact group can be extremely amenable, because, by a theorem of Veech [83], every such group admits a free G-flow (i.e., a flow for which g · x = x ⇒ g = 1 G). Nontriviality of the universal minimal flow for locally compact groups also follows from the earlier results of Granirer-Lau …