Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

THE CANTOR SET OF LINEAR ORDERS ON ℕ IS THE UNIVERSAL MINIMAL S_{∞}-SYSTEM

ELI GLASNER

Abstract. Each topological group G admits a unique universal minimal dynamical system (M(G), G). When G is a non-compact locally compact group the phase space M(G) of this universal system is nonmetrizable. There are however topological groups for which M(G) is the trivial one point system (extremely amenable groups), as well as topological groups G for which M(G) is a metrizable space and for which there is an explicit description of the dynamical system (M(G), G). One such group is the topological group S_{∞} of all permutations of the integers ℤ, with the topology of pointwise convergence. We show that (M(S_{∞}), S_{∞}) is a symbolic dynamical system (hence in particular M(S_{∞}) is a Cantor set), and give a full description of all its symbolic factors. Among other facts we show that (M(G), G) (and hence also every minimal S_{∞}) has the structure of a two-to-one group extension of proximal system and that it is uniquely ergodic.

This is a summary of a talk given at the Prague Topological Symposium of 2001 in which I described results obtained in a joint paper with B. Weiss. The paper is going to appear soon in GAFA [4].

Given a topological group G and a compact Hausdorff space X, a dynamical system (X, G) is a jointly continuous action of G on X. If (Y, G) is a second dynamical system then a continuous onto map π : (X, G) → (Y, G) which intertwines the G actions is called a homomorphism. The dynamical system (X, G) is point transitive if there exists a point x_{0} ∈ X whose orbit Gx_{0} is dense in X. (X, G) is minimal if every orbit is dense. It can be easily shown that there exists a unique (up to isomorphism of dynamical

2000 Mathematics Subject Classification. 22A05, 22A10, 54H20. Key words and phrases. dynamical systems, universal minimal systems. The author was an invited speaker at the Ninth Prague Topological Symposium. This is a summary article. The results in this article will be treated fully in an article,

written jointly with B. Weiss, to be published in Geometric and Functional Analysis (GAFA).

Eli Glasner, The Cantor set of linear orders on ℕ is the universal minimal S_{∞}-system, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 119–123, Topology Atlas, Toronto, 2002: arXiv:math.DS/0204126.

G-systems) universal point transitive G-system (L, G). One way of presenting this universal object is via the Gelfand space of the C^{∗}-algebra L_{l}(G) of left uniformly ℂ-valued continuous functions on G. From the existence of (L, G) one easily deduces the existence of a universal minimal dynamical system; i.e. a system (M(G), G) such that for every minimal system (X, G) there exists a homomorphism π : (M(G), G) → (X, G). Ellis’ theory shows that up to isomorphism this universal minimal dynamical system is unique, see e.g. [1].

The existence of uncountably many characters of the discrete group ℤ already shows that the phase space M(ℤ) is non-metrizable. In fact one can show that M(G) is non-metrizable whenever G is non-compact locally compact group.

A topological group G has the fixed point on compacta property (f.p.c.) (or is extremely amenable) if whenever it acts continuously on a compact space, it has a fixed point. Thus the group G has the f.p.c. property iff its universal minimal dynamical system is the trivial one point system.

A triple (X, d, μ), where (X, d) is a metric space and μ a probability measure on X, is called an mm-space. For A ⊆ X, μ(A) ≥ 1/2, and ϵ > 0 let A_{ϵ} be the set of all points whose distance from A is at most ϵ.

A family of mm spaces (X_{n}, d_{n}, μ_{n}) is called a Lévy family if for every ϵ, α_{n}(ϵ) → 0, where α(ϵ) = 1 − inf{μ(A_{ϵ}) : A ⊆ X, μ(A) ≥ 1/2}. When a Polish group (G, d) contains an increasing sequence of compact subgroups {G_{n} : n ∈ ℕ} whose union is dense in G and such that with respect to the corresponding sequence of Haar measures μ_{n}, the family (G_{n}, d, μ_{n}) forms a Lévy family, then G is called a Lévy group.

In [5] Gromov and Milman prove that every Lévy group G has the f.p.c. property. Many of the examples presently known of extremely amenable groups are obtained via this theorem. There are however other methods of obtaining such groups. Here is a partial list:

(1) The unitary group U(∞) = ∪_{n=1}^{∞}U(n) with the uniform operator topology (Gromov-Milman, [5]).

(2) The monothetic Polish group L_{m}(I, S^{1}), consisting of all (classes) of measurable maps from the unit interval I into the circle group S^{1} with the topology of convergence in measure induced by, say, Lebesgue measure on I (Glasner, [3]; Furstenberg-Weiss). More generally, L_{m}(I, G), where G is any locally compact amenable group (Pestov, [8]).

(3) The group of measurable automorphisms Aut(X, μ) of a standard sigma-finite measure space (X, μ), with respect to the weak topology (Giordano-Pestov [2]).

(4) Using Ramsey’s theorem, Pestov has shown that the group Aut(ℚ, <), of order automorphism of the rational numbers with pointwise convergence topology, is extremely amenable, [6].

UNIVERSAL MINIMAL S_{∞}-SYSTEM 121

Thus, as we have seen, the universal minimal system (M(G), G) corresponding to a non-compact G is usually non-metrizable but can be, in some cases, trivial. Are there non-compact topological groups for which M(G) is metrizable but non-trivial? The first such example was pointed out by Pestov [6] who used claim 4 above to show that the universal minimal dynamical system of the group G of orientation-preserving homeomorphisms of the circle coincides with the natural action of G on S^{1}.

In [9] V. Uspenskij shows that the action of a topological group G on its universal minimal system M(G) is never 3-transitive. As a direct corollary he shows that for manifolds X of dimension > 1 (as well as for X = Q, the Hilbert cube) the corresponding group G of orientation preserving homeomorphisms, (M(G), G) does not coincide with the natural action of G on X.

Let S_{∞} be the group of all permutations of the integers ℤ. With respect to the topology of pointwise convergence on ℤ, S_{∞} is a Polish topological group. The subgroup S_{0} ⊂ S_{∞} consisting of the permutations which fix all but a finite set in ℤ is an amenable dense subgroup (being the union of an increasing sequence of finite groups) and therefore S_{∞} is amenable as well.

In [5] Gromov and Milman conjectured, in view of the concentration of measure on S_{n} with respect to Hamming distance, that S_{∞} has the f.p.c. property. In [6] and [7] V. Pestov has shown that, on the contrary, S_{∞} acts effectively on M(S_{∞}) and that, in fact, there is no Hausdorff topology making S_{0} a topological group with the f.p.c. property. He as well as A. Kechris (in private communication) asked for explicit examples of S_{∞}-minimal systems.

The main result of our work [4] is the fact that the universal minimal system (M(S_{∞}), S_{∞}) is a metrizable system, in fact a system whose phase space is the Cantor set. We also give in this work an explicit description of (M(S_{∞}), S_{∞}) as a “symbolic” dynamical system and exhibit explicit formulas for all of its symbolic factors. Let me now describe these results in more details.

For every integer k ≥ 2 let

ℤ_{∗}^{k} = {(i_{1}, i_{2}, …, i_{k}) ∈ ℤ^{k} : i_{1}, i_{2}, …, i_{k} are distinct elements of ℤ},

and set Ω^{k} = {1, −1}^{ℤ_{∗}^{k}}. Consider the dynamical system (Ω^{k}, S_{∞}), where for α ∈ S_{∞} and ω ∈ Ω^{k} we let

(αω)(i_{1}, i_{2}, …, i_{k}) = ω(α^{−1}i_{1}, α^{−1}i_{2}, …, α^{−1}i_{k}).

Let Ω_{alt}^{k} ⊂ Ω^{k} consist of all the alternating configurations, that is those elements ω ∈ Ω^{k} satisfying

ω(σ(i_{1}), σ(i_{2}), …, σ(i_{k})) = sgn(σ)ω(i_{1}, i_{2}, …, i_{k}),

for all σ ∈ S_{k} and (i_{1}, i_{2}, …, i_{k}) ∈ ℤ_{∗}^{k}. Clearly Ω_{alt}^{k}is a closed and S_{∞}- invariant subset of Ω^{k}.

A configuration ω ∈ Ω^{2} determines a linear order on ℤ if it is alternating, and satisfies the conditions:

ω(m, n) = 1 ∧ ω(n, l) = 1 ⇒ ω(m, l) = 1.

Let <_{ω} be the corresponding linear order on ℤ, where m <_{ω} n iff ω(m, n) = 1. Let X = Ω_{lo}^{2} be the subset of Ω^{2} consisting of all the configurations which determine a linear order. The correspondence ω ⟷ <_{ω} is a surjective bijection between Ω_{lo}^{2} and the collection of linear orders on ℤ. Clearly X is a closed S_{∞}-invariant set and using Ramsey’s theorem we shall show that (X, S_{∞}) is a minimal system.

Say that a configuration ω ∈ Ω^{3} is determined by a circular order if there exists a sequence {z_{m} : m ∈ ℤ} ⊂ S^{1} with m ≠ n ⇒ z_{m} ≠ z_{n} such that: ω(l, m, n) = 1 for (l, m, n) ∈ ℤ_{∗}^{3} iff the directed arc in S^{1} defined by the ordered triple (z_{l}, z_{m}, z_{n}) is oriented in the positive direction. Let Y = Ω_{c}^{3} ⊂ Ω_{alt}^{3} denote the collection of all the configurations in Ω^{3} which are determined by a circular order. It follows that the set Y = Ω_{c}^{3} is closed and invariant and using Ramsey’s theorem one can show that it is minimal.

If we go now to Ω_{alt}^{4}, can one find a sequence of points {z_{n}} on the sphere S^{2} in general position such that the tetrahedron defined by any four points z_{n_{1}}, z_{n_{2}}, z_{n_{3}}, z_{n_{4}} has positive orientation when n_{1} < n_{2} < n_{3} < n_{4}?

Starting with any sequence {z_{n}} ⊂ S^{2} in general position one can use Ramsey’s theorem to find a subsequence with the required property. Another way to see this is to use the ‘moment curve’

t ↦ (t, t^{2}, t^{3}).

Again it turns out that the orbit closure in Ω_{alt}^{4} which is determined by such a sequence forms a minimal dynamical system.

It now seems as if going up to Ω_{alt}^{k} with larger and larger k’s we encounter more and more complicated minimal systems. However, as we show in [4], this is not the case and the entire story is already encoded in the simplest symbolic dynamical system Ω_{lo}^{2}.

Theorem 1. Ω_{lo}^{2} is the universal minimal S_{∞}-system.

The fact that the topology on S_{∞} is zero-dimensional, and in fact given by a sequence of clopen subgroups, enables us to reduce this theorem to the following one.

Theorem 2. Every minimal subsystem Σ of the system (Ω^{k}, S_{∞}) is a factor of the minimal system (Ω_{lo}^{2}, S_{∞}).

Finally let me mention two more facts concerning the system (M(S_{∞}), S_{∞}).

Theorem 3. The universal minimal system (Ω_{lo}^{2}, S_{∞}) has the structure of a two-to-one group extension of a proximal system.

Theorem 4. The universal minimal system (Ω_{lo}^{2}, S_{∞}) is uniquely ergodic and therefore so is every minimal S_{∞}-system.

References

[1] Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. MR 42 #2463

[2] Thierry. Giordano and Vladimir G. Pestov, Some extremely amenable groups, To appear in C. R. Acad. Sci. Paris Sér. I Math. arXiv:math.GR/0109138

http://arxiv.org/abs/math.GR/0109138, 2002.

[3] Eli Glasner, On minimal actions of Polish groups, Topology Appl. 85 (1998), no. 1-3, 119–125, 8th Prague Topological Symposium on General Topology and Its Relations to Modern Analysis and Algebra (1996). MR 99c:54057

[4] Eli Glasner and Benjamin Weiss, Minimal actions of the group 𝕊(ℤ) of permutations of the integers, To appear in Geom. Funct. Anal. (GAFA), 2002.

[5] M. Gromov and V. D. Milman, A topological application of the isoperimetric inequality, Amer. J. Math. 105 (1983), no. 4, 843–854. MR 84k:28012

[6] Vladimir G. Pestov, On free actions, minimal flows, and a problem by Ellis, Trans. Amer. Math. Soc. 350 (1998), no. 10, 4149–4165. MR 99b:54069

[7] , Amenable representations and dynamics of the unit sphere in an infinitedimensional Hilbert space, Geom. Funct. Anal. 10 (2000), no. 5, 1171–1201. MR 2001m:22012

[8] , Ramsey-milman phenomenon, urysohn metric spaces, and extremely amenable groups, Israel J. Math. 127 (2002), 317–358, arXiv:math.FA/0004010

http://www.arxiv.org/abs/math.FA/0004010.

[9] V. V. Uspenskij, On universal minimal compact g-spaces, Topology Proceedings 25 (2000), no. Spring, 301–308, arXiv:math.GN/0006081

http://at.yorku.ca/b/a/a/k/56.htm.

Department of Mathematics, Tel Aviv University, Ramat Aviv, Israel

E-mail address: glasner@math.tau.ac.il