Iterated joining means

Using a simple dynamical system generated by means M and N which are considered on adjacent intervals, we show how to find their joints, that is means extending both M and N. The procedure of joining is a local version of that presented in Jarczyk (Publ. Math. Debr. 91:235–246, 2017). Among joints are those semiconjugating some functions defined by the use of the so-called marginal functions of M and N.


Introduction
Given any interval I a function L : I × I → I is called a mean on I if min {x, y} ≤ L(x, y) ≤ max {x, y} , x,y ∈ I.
It is said to be strict if both the inequalities are sharp whenever x = y. Having an interior point ξ of I we split I into two intervals: (1.1) The present paper is, in a sense, a continuation of [1,2] where the following problem was studied: For any means M and N on I ξ and ξ I, respectively, find their common extension L to a mean on I. Any such L, denoted also by M ⊕ N , is called a joint of M and N.
A first trial to answer that question was given in the paper [1], whereas a more comprehensive description of possible solutions of the problem was presented in [2]. In both papers the authors were interested only in joints of M and N , which have all the values strictly depending on the values of M and N . This is done there by using the marginal functions h 1 , h 1 : I → I defined by In the next section we summarize some results of [2] underlying the global approach to the problem of joining means. The remaining Sects. 3-5 contain main outcomes of the present paper, where more general local notions and iteration give a wider possibility of joining means. As a conclusion we observe that some special limit joints of means M and N semiconjugate functions naturally coming from their marginal functions h 1 and h 2 .

Joiners and marginal joints
In what follows, besides intervals (1.1), we use others: Moreover, given any functions f : I ξ → I ξ and g : ξ I → ξ I, satisfying f (ξ) = g(ξ), we define the sum f ∪ g : I → I by Fix means M and N on I ξ and ξ I, respectively, consider their marginal functions h 1 and h 2 , and denote by h 1 × h 2 their product. Thus h 1 × h 2 : The notion of a joiner, recalled below, is basic for the method of joining means presented in [2]. We say that a multifunction Notice that each joiner has non-empty values.
There are a lot of joiners. Some of them are trivial, for instance all those K K K with values containing ξ. Two examples of less trivial but single-valued joiners originate in the paper [1] (for details see [2, Examples 1 and 2]). Example 4.1 below provides a description of a joiner with values which are not singletons. A simple condition sufficient for a multifunction to be a joiner can be found in Theorem 4.1.
Another notion, fundamental to the idea of joining means, is that of a marginal joint of means. The following result (see [2, Theorem 1]) serves as a starting point for its definition.
are non-empty and its every selection is a mean on the interval I, extending both the means M and N .
For an extension of Theorem 2.1 to local joiners see Theorem 4.2. Any mean

A dynamical system induced by a pair of means
As previously let h 1 and h 2 be marginal functions for means M and N given on I ξ and ξ I, respectively. Joiners considered in the previous section are of global type: they have to be defined on the whole set ( . However, it would be desirable and nice to join means making use of some local joiners, defined only in some small sets, for instance in neighbourhoods of the point (ξ, ξ). One possible idea to fulfil that requirement is to use a simple dynamical system presented below, connected with the means M and N .
The product h 1 × h 2 maps the square I × I into itself, so it can be iterated. To study properties of sequences of its iterates we begin with the following observation. Clearly, To obtain further properties of the sequence ((h 1 × h 2 ) n ) n∈N of iterates of h 1 × h 2 we need a little bit stronger assumptions.
Proof. Assertion (i) follows directly from the compactness of I, the continuity of h 1 , h 2 , and the obvious inclusions h 1 (I) ⊂ I and h 2 (I) ⊂ I.
To prove (ii) we focus on the set ∞ n=1 h n 1 (I). Observe that it is a compact interval containing all fixed points of h 1 and assume, for instance, that x 0 > x + 1 . Define the function H : I ξ → I ξ by Clearly, H is continuous and, by (3.1) and the definition of x + 1 , Thus, by [3, Theorem 1.24], the sequence (H n (x)) n∈N converges to x + 1 uniformly on I ∩ x + 1 , +∞ . Since h 1 ( ξ I) ⊂ ξ I and x 0 > ξ it follows that Take any sequence (x n ) n∈N of points of I • ξ such that x 0 = H n (x n ) for all n ∈ N. Then x + 1 < x 0 ≤ x n , n ∈ N, and, in view of the uniform convergence of (H n | I∩(x + 1 ,+∞) ) n∈N to x + 1 , we see that Analogously we show the second equality and complete the proof of (ii).
To see (iii) we argue in a standard way. Take any open set G ⊂ R 2 containing and suppose to the contrary that C n := (h 1 × h 2 ) n (I × I)\G = ∅ for all n ∈ N. Then {C n } n∈N is a family of non-empty closed subsets of the compact space I × I, with the finite intersection property, and thus ∞ n=1 C n = ∅. But this is impossible as  Proof. It is enough to observe that now x − 1 = x + 1 = x − 2 = x + 2 = ξ and to apply Theorem 3.1.

Local joiners and iterated joints
Now we are in a position to define a local joiner. Let h 1 and h 2 be the marginal functions for means M and N given on I ξ and ξ I, respectively, and let p ∈ N. We say that a multifunction K K K, defined on a subset of I × I which contains the set (  and Notice that the notion of 1-joiner coincides, in fact, with that of a joiner introduced in Sect. 2. Any p-joiner, where p ∈ N, is called a local joiner of the pair (M, N ).
There is a lot of local joiners for the pair (M, N ). Some of them are trivial. This is the case, for instance, if all the values of K K K contain ξ. A less trivial local joiner is provided by the following example, where K(x, y) contains both x and y for all (x, y) from the domain of K K K.
Analogously one can show that for all (x, y) ∈ ξ I • × I • ξ . It follows that the formulas , define a p-joiner of the pair (M, N ). Observe that u, v ∈ K K K(u, v) for all (u, v) ∈ (h 1 × h 2 ) p (I • ξ × ξ I • ∪ ξ I • × I • ξ ). Assuming additionally that h 1 and h 2 are continuous and increasing (not necessarily strictly) we have The following result gives a simple condition sufficient for a multifunction to be a local joiner for a given pair of means.