The topological Rohlin property and topological entropy

For a compact metric space X let G = H(X) denote the group of self homeomorphisms with the topology of uniform convergence. The group G acts on itself by conjugation and we say that X satisfies the topological Rohlin property if this action has dense orbits. We show that the Hilbert cube, the Cantor set and, with a slight modification, also even dimensional spheres, satisfy this property. We also show that zero entropy is generic for homeomorphisms of the Cantor set, whereas it is infinite entropy which is generic for homeomorphisms of cubes of dimension d ≥ 2 and the Hilbert cube. 0. Introduction. Let (X,X ,μ, T) be an aperiodic probability measure preserving system with μ nonatomic. Given ! > 0 and a positive integer n, Rohlin’s lemma tells us that there is a measurable subset A ⊂ X such that A, TA, T2A, . . . , Tn−1A are disjoint and cover X up to a set of measure less than !. This simple lemma is an essential tool in ergodic theory. It is used in one way or another in most aspects of this theory. One well-known consequence of it is the following. THEOREM. For a nonatomic probability space (X,X ,μ) let G be the Polish group of measure preserving transformations with a measurable inverse, equipped with the weak topology. Then the action of the group G on itself by conjugation is topologically transitive; i.e., there exists a transformation T ∈ G such that the set {STS−1: S ∈ G} is dense in G. One can consider a more general situation where a (say countable discrete) group Γ acts by measure preserving transformations on a probability space (X,X ,μ). Again the space A = AΓ of all such Γ-actions can be endowed with the weak topology, making it a Polish space, and the group G of all bi-measurepreserving-transformations of (X,X ,μ), acts on A by conjugation. In [GK] the following definition was introduced. Say that the group Γ has the Rohlin property if the action of G on AΓ is topologically transitive. It is observed there that every amenable Γ has the Rohlin property, and the question which groups have the Rohlin property is raised. (See [GK] for more details.) In the present work we are dealing with an analogous question in the topological context. For a compact metric space X, denote the group of self Manuscript received June 1, 2000. American Journal of Mathematics 123 (2001), 1055–1070. 1055 1056 ELI GLASNER AND BENJAMIN WEISS homeomorphisms of X by G = H(X). With the topology of uniform convergence, G is a Polish topological group. We say that a Polish topological group G has the topological Rohlin property (or just the Rohlin property) when it acts transitively on itself by conjugation. We say that the space X has the Rohlin property when G = H(X) has the Rohlin property; i.e. H(X) is the closure of a single conjugacy class. Which compact metric spaces possess the Rohlin property? We show that the Hilbert cube and the Cantor set have it. For some connected spaces like spheres the existence of orientation of a homeomorphism, which is clearly preserved under conjugation, means that H(Sd) cannot have the Rohlin property; therefore we say that a sphere satisfies the Rohlin property when the group H0(S)—the connected component of the identity in H(Sd)—has the Rohlin property. With this definition we show that even dimensional spheres have the Rohlin property. On the other hand it appears that for general compact manifolds of positive finite dimension the answer is rather different. For circle homeomorphisms, Poincaré’s rotation number, τ : H+(S1) → R/Z, h $→ τ (h), where H+(S1) = H0(S) is the subgroup of index 2 of orientation preserving homeomorphisms, is a continuous conjugation invariant and thus there are at least a continuum of different closed disjoint conjugation invariant subsets. We refer the reader to the recent paper [AHK], by E. Akin, M. Hurley and J. Kennedy, for a detailed discussion of circle homeomorphisms. Their main result (on the circle) can be briefly formulated by saying that the circle has the local Rohlin property, where a space X has the local Rohlin property if H(X) contains an open dense subset which is the union of interior of conjugacy class closures. More precisely they show that for a rational number c, the set τ−1(c) has a nonempty interior in H+(S1) and that in each such set τ−1(c) there is a—necessarily unique—residual H+(S1) conjugacy class. On the other hand for irrational rotation numbers we have the following information. Denote by T the set of topologically transitive homeomorphisms of S1, then T is a Gδ subset of H+(S1) on which τ takes irrational values and for an irrational number c the set T ∩ τ−1(c) is the conjugacy class of the “rigid” rotation hc. It is also easy to see that no odd dimensional sphere has the Rohlin property. For the proof it suffices to note that there are orientation preserving homeomorphisms with an attracting fixed point—hence all small perturbations have a fixed point, while there are orientation preserving homeomorphisms with no fixed points—and any small perturbation will not have one either. The motivation for the definition of the Rohlin property came from the work [GK]. The Hilbert cube case was done during the special year in ergodic theory at the Institute for Advanced Studies of the Hebrew University in Jerusalem, 1996–97. The question regarding the Cantor set was raised recently by J. King and was answered independently by E. Akin, [AHK]. A related problem is the question: what is the topological entropy of the typical homeomorphism in H(X)? The machinery we develop for dealing with THE TOPOLOGICAL ROHLIN PROPERTY AND TOPOLOGICAL ENTROPY 1057 the Rohlin property enables us to answer the entropy problem as follows. For the Hilbert cube and spheres Sd, d ≥ 2, the set of homeomorphisms with infinite entropy is residual while for the Cantor set it is the set of zero entropy which is a dense Gδ subset of H(X). The Hilbert cube is dealt with in Section 1, the Cantor set in Section 2 and in Section 3 we consider finite dimensional cubes and spheres. Acknowledgments. We wish to thank E. Akin for his helpful comments and for supplying information concerning the Annulus conjecture. 1. The Hilbert cube. Let X be a compact metric space and let H(X) be the group of self homeomorphisms of X equipped with the metric: ρ(S, T) = sup x∈X {d(Sx, Tx)} + sup x∈X {d(S−1x, T−1x)}. With this metric H(X) is a Polish topological group. This metric though, is not in general a right invariant metric. However, every second countable topological group admits a complete right invariant metric (see e.g. [HR], Theorem 8.2), and for convenience we shall fix such an invariant metric D on H(X): D(SR, TR) = D(S, T), ∀R, S, T ∈ H(X). We say that X has the Rohlin property if the action of the group H(X) on itself by conjugation is topologically transitive; i.e if there exists a homeomorphism T ∈ H(X) such that the set {STS−1: S ∈ H(X)} is dense in H(X). From general considerations it then follows that this property holds for a dense Gδ subset of H(X). THEOREM 1.1. The Hilbert cube Q = [ − 1, 1] = J has the Rohlin property. Proof. For an element R of H(Q) and a positive ! put E(R, !) = {T ∈ H(Q): ∃S ∈ H(Q), d(S−1TS, R) < !}. Clearly E(R, !) is an open subset of H(Q). We will show that it is also dense in H(Q). If {Ri}i=1 is a dense sequence in H(Q) then by Baire’s category theorem the Gδ subset