The Birkhoff Center and Analytic Sets

Abstract

It is shown here how G. D. Birkhoff's notion of the center of a homeomorphism or flow naturally gives rise to an analytic set in a product space. It is shown that for a wide class of spaces this set is not a Borel set. Let X be a locally compact separable metric space with complete metric d and let H(X) be the space of autohomeomorphisms of X. The space H(X) has a topology under which it is a complete separable metric group [6, 9]. For a wide class of Xs, it is known that this topology is unique [7]. This topology may be briefly described as follows. Let X* = X U {oo} be the one point compactification of X and consider the space M = M(X*, X*) of all continuous maps of X* into X* provided with the compact open topology [9]. In this topology, M is a Polish space: M is separable and possesses a complete metric compatible with this topology. Identify H(X) with F = {(f, g) E M x M:fg = gf = idx* and f(oo) = oo}. Since F is closed in M x M, F is also a Polish space. We consider H(X) to have this topology. If h E H(X) and Y is an h-invariant subset of X, then a point y E Y is said to be nonwandering with respect to Y provided there is an increasing sequence of positive integers n1, n2, n3, . . . and points yp E Y, p = 1, 2, 3, . . . such that the sequence hnp(yp) converges to y. Let Rh(Y) = {y E Y:y is nonwandering with respect to Y}. If Y is a closed h-invariant set, then Rh(Y) is also closed and h-invariant. Set R(X) = X and by recursion, for each ordinal ox, Ro '(X) = Rh(Ro'(X)) and, if X is a limit ordinal, Rx(X) = n,a<x R-(X). Since this central sequence {RO'(X)} forms a decreasing transfinite sequence of closed sets in X, there is a least countable ordinal 8 = 8(h) such that Rh'(X) = R'(X). This ordinal is called the depth of h and R'(X) = Rh(X) is called the center of h. Of course, Rh(X) is the closure of the set of all h-recurrent points.