Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 17, 2005, 43 – 46

© R.N. Gumerov

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, $n\ge 2$, on a topological space $X$ is a continuous mapping $\mu$ from the Cartesian product of $n$ copies of $X$ into $X$ such that $\mu \left(x,x,\dots ,x\right)=x$ and $\mu \left(x_1,x_2,\dots ,x_n\right)=\mu \left(x_{\sigma \left(1\right)},x_{\sigma \left(2\right)},\dots ,x_{\sigma \left(n\right)}\right)$ for all $x,x_1,x_2,\dots ,x_n\in X$ and any permutation $\sigma$ of the set $\left\{1,2,\dots ,n\right\}$.

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 $n$-mean 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 $H^1\left(G,\mathbb{\mathbb{Z}}\right)$ with the integers $\mathbb{\mathbb{Z}}$ 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 $h\in H$, there exists an element $g\in H$ such that $h=ng$.

Let $N=\left(n_1,n_2,\dots \right)$ be a sequence of integers that are greater than $1$. The solenoid ${\Sigma }_N$ is defined as the inverse limit of the inverse sequence  $\left\{\right.X_k,f_k^{k+1},\mathbb{\mathbb{N}}\left\}\right.$, where $\mathbb{\mathbb{N}}$ is the set of all positive integers and for each $k\in \mathbb{\mathbb{N}}$  the factor space  $X_k$  is the unit circle  ${\mathbb{S}}^1$  in the complex plane and the bonding mapping $f_k^{k+1}$ is the $n_k$-fold covering mapping ${\mathbb{S}}^1\to {\mathbb{S}}^1:z\mapsto z^{n_k}$. The solenoid is a compact connected Abelian group under the coordinatewise multiplication with the identity $e=\left(1,1,\dots \right)$. In case $N=\left(2,2,\dots \right)$ the solenoid ${\Sigma }_N$ is said to be dyadic.

The exponential covering mapping from the reals  $\mathbb{ℝ}$  onto  ${\mathbb{S}}^1$  induces a one-to-one continuous homomorphism  $\theta :\mathbb{ℝ}\to {\nprec }_N$  defined by

which is not a topological embedding. The image of the homomorphism $\theta$ is the arc component of  ${\Sigma }_N$  containing the identity $e$ (see [6, $\S$ 5]).

The Pontryagin dual of  ${\Sigma }_N$  is isomorphic (see [7, (25.3)]) to the discrete additive group of rationals  ${\mathbb{\mathbb{Q}}}_N$  generated by the set

It is easy to see that the group ${\mathbb{\mathbb{Q}}}_N$ 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 ${\Sigma }_N$ 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 ${\Sigma }_N$ admits an $n$-mean and $p\ge 2$ is a prime factor of the integer $n$. It follows immediately from the definition of an $n$-mean that the solenoid ${\Sigma }_N$ admits a $p$-mean as well. We denote by $\mu$ a $p$-mean on ${\Sigma }_N$.

According to Theorem 2, there exists a continuous homomorphism

which is homotopic to $\mu$.

Choose any $g\in {\Sigma }_N$. The points $g$ and $\phi \left(g,g,\dots ,g\right)$ lie in the same arc component $\Gamma$ of ${\Sigma }_N$. It follows then from the equality

that  ${\left(\mu \left(g,e,\dots ,e\right)\right)}^p\in \Gamma$. Consequently, we have $\phi \left(g^p,e,\dots ,e\right)\in \Gamma$. This implies that $\mu \left(g^p,e,\dots ,e\right)\in \Gamma$. In other words, for each $g\in {\Sigma }_N$, both points $g$ and $\mu \left(g^p,e,\dots ,e\right)$ are contained in the same arc component of the space ${\Sigma }_N$.

To obtain a contradiction we suppose now that there exists an integer $k\in \mathbb{\mathbb{N}}$ such that $n_j$ is not a multiple of $p$ for each $j\ge k$. Since ${\Sigma }_N$ and ${\Sigma }_{\left(n_k,n_{k+1},\dots \right)}$ are homeomorphic we can assume that for every $j\in \mathbb{\mathbb{N}}$ the prime $p$ is not a divisor of the integer $n_j$.

Now we shall construct an element $g\in {\Sigma }_N$ 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 $\xi =exp\left(i\frac{2\pi }{p}\right)$. For each term $n_j$ of the sequence $N$ we consider the homomorphism ${\psi }_{n_j}:\sqrt[p]{1}\to \sqrt[p]{1}$ defined by ${\psi }_{n_j}\left(z\right)=z^{n_j},z\in \sqrt[p]{1}$. Since the integers $p$ and $n_j$ are relatively prime the mapping ${\psi }_{n_j}$ is a bijection. Denote by ${\phi }_{n_j}:\sqrt[p]{1}\to \sqrt[p]{1}$ the inverse of ${\psi }_{n_j}$. We have ${\left({\phi }_{n_j}\left(z\right)\right)}^{n_j}=z$ for each $z\in \sqrt[p]{1}$. So that the sequence

is an element of  ${\Sigma }_N$ and $g^p=e$. Therefore we get $\mu \left(g^p,e,\dots ,e\right)\in \theta \left(\mathbb{ℝ}\right)$.

On the other hand, it is obvious that the point $g$ does not lie in $\theta \left(\mathbb{ℝ}\right)$ (see also Remark below). Thus the points $g$ and $\mu \left(g^p,e,\dots ,e\right)$ belong to distinct arc components of the space ${\Sigma }_N$. 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  ${\Sigma }_N$  is homeomorphic to ${\Sigma }_M$, where $M=\left(m_1,m_2,\dots \right)$ with $m_j=n{k}_j,k_j\in \mathbb{\mathbb{N}}$, for all $j\in \mathbb{\mathbb{N}}$. It is straightforward to check that the mapping

determined by the formula

is an $n$-mean on the solenoid ${\Sigma }_M$. This completes the proof of Theorem 1.

Remark. It is interesting to note that any non-trivial continuous self-homomorphism of the solenoid ${\Sigma }_N$ 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 finite-sheeted connected coverings of solenoids (see, e.g., [9, Theorem 2]), one can easily obtain the following theorem.

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

References

Department of Mechanics and Mathematics, Kazan State University, Kremlevskaya 18, Kazan, 420008, Russian Federation

E-mail address: renat.gumerov@ksu.ru