Embeddability of pairs of weighted quasi-arithmetic means into a semiflow

We determine the form of all semiflows of pairs of weighted quasi-arithmetic means, those over positive dyadic numbers as well as the continuous ones. Then the iterability of such pairs is characterized: necessary and sufficient conditions for a given pair of weighted quasi-arithmetic means to be embeddable into a continuous semiflow are given. In particular, it turns out that surprisingly the existence of a square iterative root in the class of such pairs implies the embeddability.


Introduction
One crucial problem of dynamical system is the embeddability of an individual function into a flow or, more generally, into a semiflow. Given an arbitrary set X and a subgroup T of the additive group of reals [subsemigroup T of the additive semigroup of positive reals] any function F : X × T → X, satisfying the celebrated translation equation is called a T -flow [T -semiflow ] in X. In the case when X is endowed with a topology a flow F : X × R → X [semiflow F : X × (0, +∞) → X] is said to be continuous if the function F (x, ·) is continuous for each x ∈ X. In iteration theory continuous flows and semiflows are usually called continuous iteration groups and semigroups, respectively. A function f : X → X is said to be embeddable into a T -flow [T -semiflow ] if there exists a flow [semiflow] F : X × T → X such that f = F (·, 1). Embeddability problems have been discussed by many authors in different settings. The following ones can serve as examples of such research. In particular, in 1968 Karlin and McGregor published two papers on this: [7] concerning the embeddability of branching processes and [8] dealing with such a problem for analytic functions with two fixed points. Embeddability of homeomorphisms into differential flows was studied by Ping Fun Lam [12] in 1976. A number of papers devoted to different variants of the embedding problem was written by Zdun. He answered, among others, the questions of how to embed continuous strictly monotonic functions defined on an interval [15], homeomorphisms of the circle [16], two commuting functions [17] and, jointly with Solarz, diffeomorphisms of the plane in a regular iteration semigroup [20]. Also some problems close to embedding were considered, like approximative embedding (see [18] by Zdun) and embeddability of homeomorphisms of the circle into set-valued flows (see [9] by Krassowska and Zdun). Embedding problems have been discussed also in a number of monographs and surveys, e.g. [10] by Kuczma,[14] by Targonski, [11] by Kuczma, Choczewski and Ger, [2] by Belitskii and Tkachenko, [19] by Zdun and Solarz. In 2010 the third present author and Matkowski considered the embeddability problem for pairs of homogeneous symmetric means; see [6]. Given any real interval I a two-variable mean M on I is any function M : for all t ∈ (0, +∞) or for all t ∈ R, respectively. Here we study the embeddability of pairs of weighted quasi-arithmetic means, that is means A f p of the form where f : I → R is any continuous strictly monotonic function and p ∈ (0, 1  [5]). So the research reported here substantially differs from that presented in [6].
Denote by CM(I) the class of continuous strictly monotonic functions mapping the interval I into R. We say that functions f, g ∈ CM(I) are equivalent if there are numbers a ∈ R \ {0} and b ∈ R such that then we write f ∼ g. Clearly, ∼ is an equivalence relation in the set CM(I).
Using this notion one can formulate the following result solving the so-called equality problem for weighted quasi-arithmetic means (see [1, Sec. 6.4.3, Theorem 2] also [13] by Maksa and Páles; for quasi-arithmetic means the answer was known already in 1934 and can be found in the book [5]).
and only if f ∼ g and p = q.

Posing the problem
We start with the definition of iterability. Given an additive semigroup T ⊂ (0, +∞) containing 1, a set X and a family A of self-mappings of X, a function Denote by A the set of pairs of weighted quasi-arithmetic means on an interval I: . Our first aim is to find the form of all continuous semiflows of pairs from A. As a consequence we obtain the main result giving a characterization of the continuous iterability of an arbitrary pair A f p , A g q ∈ A in A. The main tool used in the proofs presented in Sects. 3, 4 and 5 is the following result originating from the paper [4]. Here Theorem B. Let f, g, ϕ, ψ ∈ CM(I) and p, q, μ, ν ∈ (0, 1). Then the pair and one of the following cases holds: (ii) (p, q) ∈ Δ \ Γ and condition (1) is satisfied.

The form of continuous semiflows of pairs of weighted quasi-arithmetic means
To prove Theorems 4 and 6, which are the main results of this section, we need some auxiliary facts. The first one reduces determining semiflows of pairs from A to solving a system of functional equations.

Proposition 1.
Let T ⊂ (0, +∞) be an additive semigroup, μ, ν : T → (0, +∞) and let f ∈ CM(I). The function F : is a semiflow if and only if the pair (μ, ν) satisfies the system of equations In the proof we use the following lemma which can be easily verified.
The next result will be derived from Theorems B and A.

Proposition 3.
If (F n ) n∈N0 is a sequence of pairs of weighted quasi-arithmetic means on I such that F n • F n = F n−1 for all n ∈ N, then there exist a function f ∈ CM(I) and numbers p, q ∈ (0, 1) such that p ≥ q and where Proof. By Theorems B and A there exist a function f ∈ CM(I), numbers p, q ∈ (0, 1) and sequences (p n ) n∈N0 and (q n ) n∈N0 of numbers from (0, 1) such that p 0 = p, q 0 = q, p n ≥ q n for all n ∈ N 0 and (3) holds true; moreover, either for every n ∈ N 0 . Suppose that the conditions in (5) do not hold for some n ∈ N 0 . Then the equalities in (6) are true, hence Consequently, p n = q n , also p n+1 = q n+1 , and thus (5) follows contrary to the supposition. This implies that the equalities in (5) hold for all n ∈ N 0 . In view of (5), where n is replaced by n − 1, we have which implies the condition Now, using the first equality in (5), we prove by induction the first part of (4) and then, applying (7), also the second one.   where the functions μ, ν : D + → (0, 1) are given by In the proof of Theorem 4 we need the following useful lemma.
Proof. For any t ∈ D + we have Hence, if s, t ∈ D + , then and To prove the remaining properties it is enough to notice that if p > q, then the function D + t −→ (p − q) t is strictly decreasing and Proof of Theorem 4. Assume that F : I 2 × D + → I 2 is of the form (8) with some function f ∈ CM(I), numbers p, q ∈ (0, 1) such that p ≥ q, and the functions μ, ν : D + → R defined by (9). Then Lemma 5 implies that μ (D + ) , ν (D + ) ⊂ (0, 1) and (μ, ν) satisfies system (2). Therefore, on account of Proposition 1, the function F is a semiflow. Now assume that F : I 2 × D + → I 2 is a semiflow. Then, by Proposition 3 applied to the sequence (F n ) n∈N0 given by F n = F (·, 1/2 n ) , we see that for some f ∈ CM(I), numbers p, q ∈ (0, 1) satisfying p ≥ q and the sequences (p n ) n∈N0 and (q n ) n∈N0 defined by (4). Therefore, according to (4), where μ, ν : D + → R are given by (9). It follows from Lemma 5 and Proposition 1 that the mapping is a semiflow, and thus, taking arbitrary numbers k ∈ N and n ∈ N 0 , we have , that is condition (8) holds.
Since D + is a dense subset of (0, +∞) we get the following characterization of continuous semiflows of pairs of weighted quasi-arithmetic means as an immediate consequence of Theorem 4. Theorem 6. Let F : I 2 × (0, +∞) → I 2 be a function such that F (·, t) is continuous for each t ∈ (0, +∞). The function F is a continuous semiflow of pairs of weighted quasi-arithmetic means if and only if there exist a function f ∈ CM(I) and numbers p, q ∈ (0, 1) such that p ≥ q and where the functions μ, ν : (0, +∞) → (0, 1) are given by (9).

Iterability of pairs of weighted quasi-arithmetic means
In the present section we formulate and prove a little bit surprising characterization of iterability of pairs A f p , A g q in the family A. Theorem 7. Let f, g ∈ CM(I) and p, q ∈ (0, 1). Then the following statements are pairwise equivalent: the pair A f p , A g q has a square iterative root being an element of the family A; (v) f ∼ g and p ≥ q.
Proof. The implications (i) ⇒ (ii), (ii) ⇒ (iii) and (iii) ⇒ (iv) are obvious, whereas (iv) ⇒ (v) follows from Theorem B. To get the implication (v) ⇒ (i) it is enough to make use of Theorems 6 and A.

Solutions of system (2)
We complete the paper with solving the system of equations (2). Notice that this is done as a byproduct of studying the embeddability problem. Theorem 8. Let μ, ν : T → (0, +∞), where T is either D + , or (0, +∞). If T = D + , then the pair (μ, ν) is a solution of system (2) if and only if the equalities in (9) hold for all t ∈ D + . If T = (0, +∞), then the pair (μ, ν) is a continuous solution of system (2) if and only if the equalities in (9) hold for all t ∈ (0, +∞).
Proof. If the pair (μ, ν) satisfies (2), then one can apply Proposition 1 and then Theorem 4 in the case T = D + and Theorem 6 when T = (0, +∞). To obtain the converse it is enough to use Lemma 5 and, if T = (0, +∞), additionally to recall the density of D + in (0, +∞) and the assumed continuity of μ and ν.
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