Lobachevskii Journal of Mathematics

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Vol. 17, 2005, 43 – 46 \begin{matrix}c \\ ◯ \end{matrix} R.N. Gumerov

R.N. Gumerov

ON THE EXISTENCE OF MEANS ON SOLENOIDS

(submitted by V. Lychagin)

Abstract. A mean on a topological space is a continuous idempotent and symmetric operation on it. A proof of a criterion for the existence of means on solenoids is given.

An n-mean, n ≥ 2, on a topological space X is a continuous mapping μ from the Cartesian product of n copies of X into X such that μ(x, x, …, x) = x and μ(x_{1}, x_{2}, …, x_{n}) = μ(x_{σ(1)}, x_{σ(2)}, …, x_{σ(n)}) for all x, x_{1}, x_{2}, …, x_{n} ∈ X and any permutation σ of the set {1, 2, …, n}.

There is a large literature concerning the problem on the existence of means on topological spaces (see, e.g., [1]—[5] and the references cited there). G. Aumann [2] showed that the circle does not admit an nmean for any n. J. Keesling [4] gave necessary and sufficient conditions for the existence of n-means on compact connected Abelian topological groups. In particular, a compact connected Abelian group G admits an n-mean if and only if the one-dimensional Čech cohomology group H^{1}(G, ℤ) with the integers ℤ as the coefficient group, or equivalently the Pontryagin dual of G, is n-divisible (see [4, Theorem 1.1]). We recall that an additive Abelian group H is said to be n-divisible provided that, for each element h ∈ H, there exists an element g ∈ H such that h = ng.

Let N = ( n_{1}, n_{2}, …) be a sequence of integers that are greater than 1. The solenoid Σ_{N} is defined as the inverse limit of the inverse sequence

Key words and phrases. n-mean, solenoid.

2000 Mathematical Subject Classification. 54H11, 54F15.

{X_{k}, f_{k}^{k+1}, ℕ}, where ℕ is the set of all positive integers and for each k ∈ ℕ the factor space X_{k} is the unit circle 𝕊^{1} in the complex plane and the bonding mapping f_{k}^{k+1} is the n_{k}-fold covering mapping 𝕊^{1} ⟶ 𝕊^{1} : z ↦ z^{n_{k}}. The solenoid is a compact connected Abelian group under the coordinatewise multiplication with the identity e = (1, 1, …). In case N = ( 2, 2, …) the solenoid Σ_{N} is said to be dyadic.

The exponential covering mapping from the reals ℝ onto 𝕊^{1} induces a one-to-one continuous homomorphism θ : ℝ → ⊀_{N} defined by

θ(α) = (exp(i2πα), exp(i\frac{2πα}{n_{1}}), exp(i\frac{2πα}{n_{1}n_{2}} ), …), α ∈ ℝ, i^{2} = −1,

which is not a topological embedding. The image of the homomorphism θ is the arc component of Σ_{N} containing the identity e (see [6, § 5]).

The Pontryagin dual of Σ_{N} is isomorphic (see [7, (25.3)]) to the discrete additive group of rationals ℚ_{N} generated by the set

\{\frac{1}{n_{1}} , \frac{1}{n_{1}n_{2}} , ⋯, \frac{1}{n_{1}n_{2} … n_{k}} , ⋯ | k ∈ ℕ\} .

It is easy to see that the group ℚ_{N} is n-divisible if and only if each prime factor of n divides infinitely many terms of the sequence N.

Thus, by the above-mentioned facts, one has the criterion for the existence of n-means on solenoids:

Theorem 1. The solenoid Σ_{N} admits an n-mean if and only if each prime factor of n divides infinitely many terms of the sequence N.

A simple proof of Theorem 1 for 2-means was given by P. Krupski in [5]. It is based on a method of B. Eckmann [3] and the following theorem of W. Scheffer ([8, Corollary 2]).

Theorem 2. Let G be a compact connected topological group, and H be a locally compact Abelian topological group. Then every continuous mapping from G into H that preserves the unit element is homotopic to exactly one continuous homomorphism from G into H, and the homotopy can be chosen to preserve the identity.

In this note a similar proof of Theorem 1 is given for arbitrary n-means. Proof of Theorem 1. Necessity. Suppose that the solenoid Σ_{N} admits

an n-mean and p ≥ 2 is a prime factor of the integer n. It follows immediately from the definition of an n-mean that the solenoid Σ_{N} admits a p-mean as well. We denote by μ a p-mean on Σ_{N}.

According to Theorem 2, there exists a continuous homomorphism

ϕ : Σ_{N} × Σ_{N} × … × Σ_{N} → Σ_{N},

which is homotopic to μ.

Choose any g ∈ Σ_{N}. The points g and ϕ(g, g, …, g) lie in the same arc component Γ of Σ_{N}. It follows then from the equality

ϕ(g, g, …, g) = ϕ(g, e, …, e)ϕ(e, g, e, …, e)ϕ(e, …, e, g)

that (μ(g, e, …, e))^{p} ∈ Γ. Consequently, we have ϕ(g^{p}, e, …, e) ∈ Γ. This implies that μ(g^{p}, e, …, e) ∈ Γ. In other words, for each g ∈ Σ_{N}, both points g and μ(g^{p}, e, …, e) are contained in the same arc component of the space Σ_{N}.

To obtain a contradiction we suppose now that there exists an integer k ∈ ℕ such that n_{j} is not a multiple of p for each j ≥ k. Since Σ_{N} and Σ_{(n_{k},n_{k+1},...)} are homeomorphic we can assume that for every j ∈ ℕ the prime p is not a divisor of the integer n_{j}.

Now we shall construct an element g ∈ Σ_{N} such that g^{p} = e (cf. [9, the proof of Proposition 4]). Let \sqrt[p]{1} denote the multiplicative cyclic group of all values of the p-th root of 1 generated by ξ = exp(i\frac{2π}{p}). For each term n_{j} of the sequence N we consider the homomorphism ψ_{n_{j}} : \sqrt[p]{1} → \sqrt[p]{1} defined by ψ_{n_{j}}(z) = z^{n_{j}}, z ∈ \sqrt[p]{1}. Since the integers p and n_{j} are relatively prime the mapping ψ_{n_{j}} is a bijection. Denote by ϕ_{n_{j}} : \sqrt[p]{1} → \sqrt[p]{1} the inverse of ψ_{n_{j}}. We have (ϕ_{n_{j}}(z))^{n_{j}} = z for each z ∈ \sqrt[p]{1}. So that the sequence

g = (ξ, ϕ_{n_{1}}(ξ), ϕ_{n_{2}} ∘ ϕ_{n_{1}}(ξ), …)

is an element of Σ_{N} and g^{p} = e. Therefore we get μ(g^{p}, e, …, e) ∈ θ(ℝ).

On the other hand, it is obvious that the point g does not lie in θ(ℝ) (see also Remark below). Thus the points g and μ(g^{p}, e, …, e) belong to distinct arc components of the space Σ_{N}. This contradicts the observation made above.

Sufficiency. If each prime factor of an integer n divides infinitely many terms of the sequence N, then one can readily show that the solenoid Σ_{N} is homeomorphic to Σ_{M}, where M = (m_{1}, m_{2}, …) with m_{j} = nk_{j}, k_{j} ∈ ℕ, for all j ∈ ℕ. It is straightforward to check that the mapping

μ : Σ_{M} × Σ_{M} × … × Σ_{M} → Σ_{M}

determined by the formula

μ((z_{11}, z_{12}, …), (z_{21}, z_{22}, …), …, (z_{n1}, z_{n2}, …)) =
= ((z_{12}z_{22} … z_{n2})^{k_{1}}, (z_{13}z_{23} … z_{n3})^{k_{2}}, …)

is an n-mean on the solenoid Σ_{M}. This completes the proof of Theorem 1.

46 R.N. GUMEROV

Remark. It is interesting to note that any non-trivial continuous selfhomomorphism of the solenoid Σ_{N} bijectively maps arc components onto arc components (see [10, Proposition 3]).

It is known that the problem on the existence of n-means on compact connected Abelian groups is closely related to the question of existence or nonexistence of finite-sheeted connected coverings (see, e.g., [11]). By a connected covering of a topological group we mean a covering mapping from a connected Hausdorff topological space onto a group. Using Theorem 1 and the conditions for the existence and nonexistence of finitesheeted connected coverings of solenoids (see, e.g., [9, Theorem 2]), one can easily obtain the following theorem.

Theorem 3. The solenoid Σ_{N} admits an n-mean if and only if for each prime factor p of n there is no p-fold connected covering of Σ_{N}.

References

[1] Kolmogoroff A. N., Sur la notion de la moyenne, Atti Acad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat., (6) 12 (1930), 388–391.

[2] Aumann G., Über Räume mit Mittelbildungen, Math. Ann., 119 (1944), 210–215.

[3] Eckmann B., Räume mit Mittelbildungen, Comment. Math. Helv., 28 (1954), 329–340.

[4] Keesling J., The group of homeomorphisms of a solenoid, Trans. Amer. Math. Soc., 172 (1972), 119–131.

[5] Krupski P., Means on solenoids, Proc. Amer. Math. Soc., 131 (2002), 1931–1933.

[6] McCord M. C., Inverse limit sequences with covering maps, Trans. Amer. Math. Soc., 114 (1965), 197–209.

[7] Hewitt E., Ross K. A., Abstract harmonic analysis, Vol. I, Springer–Verlag, Berlin, 1963.

[8] Scheffer W. A., Maps between topological groups that are homotopic to homomorphisms, Proc. Amer. Math. Soc., 33 (1972), 562–567.

[9] Gumerov R. N., On finite-sheeted covering mappings onto solenoids, Proc. Amer. Math. Soc. (to appear).

[10] Aarts J. M., Fokkink R. J., Mappings on the dyadic solenoid, Comment. Math. Univ. Carolinae, 2003, 44 (4), 697–699.

[11] Gumerov R. N., Means on compact groups, Trudy Lobachevskii Centre Math., Vol. 25, 97–98, International Conference, Kazan Math. Soc., 2004 (in Russian).

Department of Mechanics and Mathematics, Kazan State University, Kremlevskaya 18, Kazan, 420008, Russian Federation E-mail address: renat.gumerov@ksu.ru