A family of functions {f^t : I → I, t ∈ ℝ} such that f^t ○ f^s = f^t+s, t, s ∈ ℝ is said to be an iteration group, however a family of set-valued functions {F^t : I → 2^I, t ∈ ℝ} such that F^t ○ F^s = F^t+s, t, s ∈ ℝ is said to be a set-valued iteration group (abbreviated to s-v iteration group). The notion of an iteration semigroup of set-valued functions was introduced and studied by Smajdor [1] and then studied in some classes of set-valued functions (see e.g., [2], [3], [4], [5]). The fundamental problem in the theory of multivalued iteration semigroups is the problem of existence and regularity properties of continuous selections. In this note we considered particular set-valued iteration groups whose values are the intervals or singletons. The presented results complete and generalize some of the topics from Zdun [6]. The considered s-v iteration groups have the very irregular properties. For every such s-v iteration group {F^t : I → 2^I, t ∈ ℝ} we find a special maximal additive subgroup T ⊂ ℝ such that group {F^t : I → 2^I, t ∈ T} has several “regular” properties.

Let I = (a, b) and φ : I → ℝ be a surjection. Define the set-valued functions

The surjection φ is said to be the generating function of the family {F^t}.

Theorem 1

The family {F^t : I → 2^I} is a set-valued iteration group, i.e.,

where

Moreover, x ∉ F^t(x) for t ≠ 0.

Proof. Fix an x ∈ I. Let z ∈ F^t ○ F^s(x). Then there exists a y ∈ F^s(x) such that z ∈ F^t(y). This means that φ(y) = φ(x) + s and φ(z) = φ(y) + t, which gives that φ(z) = φ(x) + t + s. Hence z ∈ F^t+s(x). Similarly we prove the converse inclusion.                                                    □

If φ is a homeomorphism then Equation (1) defines the general form of continuous iteration groups such that F¹(x) ≠ x for x ∈ I.

If φ is non-injective then s-v iteration group generated by φ has very irregular properties and we will call this group singular. The purpose of this paper is the study of these “singularities.”

Obviously the set-valued functions F^t, t ∈ ℝ pairwise commute. This property is not transferible on the continuous selections of these set-valued mappings.

Let us assume that there exist F^u, F^v with uv∉ℚ which possess homeomorphic commuting selections f and g, that is f(x) ∈ F^u(x) and g(x) ∈ F^v(x) for x ∈ (a, b) and f ○ g = g ○ f. Then the generating function φ satisfies the equations φ(f(x)) = φ(x) + u and φ(g(x)) = φ(x) + v. Note that then f, g are iteratively incommensurable, i.e.,

where fⁿ denotes the n-th iterate of function f and f⁰ = id. Define

The set L_{f, g} does not depend on x and either this set is the interval cl I or L_{f, g} is a nowhere dense perfect set in I (see Zdun [7]). If the generating function φ is continuous at least at one point of L_{f, g} then it is continuous and it is monotonic (see [8]).

We have more

Theorem 2

If f and g are commuting iteratively incommensurable homeomorphisms, then there exist infinitely many s-v iteration groups {F^t, t ∈ ℝ} of type (1) such that f(x) ∈ F¹(x) and g(x) ∈ F^s(x) for an s ∉ ℚ, but the only one of them has a monotonic generating function φ. Then the generating function φ is continuous and φ[L_f,g] = ℝ.

The proof is a simple consequence of Theorem 2 and Corollary 1 in Zdun [8].

The family {F^t, t ∈ ℝ} is a single-valued iteration group if and only if L_{f, g} = [a, b]. Then φ is strictly monotonic (see Zdun [8]).

In this paper we consider the case where L_{f, g} ≠ [a, b], that is {F^t : t ∈ ℝ} is a proper set-valued iteration group.

In the next section we will consider the more general case.

Assume the following general hypothesis:

Then the function φ is continuous and the values of F^t are closed intervals or singletons. Denote by {I_α, α ∈ A} a family of the intervals of constancy of φ. These intervals are closed. Put

and

Note that φ|L* is strictly increasing, φ[I_α] are singletons and if I_α < I_β then φ[I_α] < φ[I_β].

It is easy to verify that the s-v iteration group {F^t : I → cc[I], t ∈ ℝ} generated by φ has the following properties.

Proposition 1

The conditions (i), (ii), (iii) characterize the interval-valued iteration groups. We have the following.

Proposition 2

If an s-v iteration group {F^t, t ∈ ℝ} satisfies conditions (i), (ii), and (iii), where {I_α, α ∈ A} is a given family of closed disjoint proper intervals, then there exists a function φ satisfying (H) such that F^t are given by the formula (1).

Proof. Define

Let x₀ ∈ I and put h(t):=Ft(x0). Note that h is a bijection from ℝ onto X. Define φ by the following way: if x ∈ I_α for an α ∈ A then φ(x):=h-1(Iα), if x ∈ L^* then φ(x): = h({x}). It is easy to see that φ is a non-decreasing surjection of I onto ℝ constant on the intervals I_α and

Since Ft ○ Fs(x0)=Ft+s(x0) we have

Hence

Let x ∈ I. Then, by (iii), there exists an s ∈ ℝ such that x ∈ h(s). Hence φ(x) ∈ φ[h(s)] = s, thus φ(x) = s. This gives that φ[F^t(x)] ⊂ φ[F^t(h(s)] = φ(x) + t, so

Since F^t(x) ⊂ φ⁻¹[φ[F^t(x)]] we have F^t(x) ⊂ φ⁻¹[φ(x) + t]. Note that φ⁻¹[φ(x) + t] is a singleton or equals to one of the intervals I_α. If F^t(x) is a singleton then, by (i), Ft(x)∉Iα for any α ∈ A. Thus φ⁻¹[φ(x) + t] is not any of the intervals I_α, so it is a singleton. If F^t(x) is an interval I_α, then φ⁻¹[φ(x + t)] must be also the same interval. This gives equality F^t(x) = φ⁻¹[φ(x + t)].                   □

Proposition 3

Let a family of set-valued function F^t be given by (1), where φ satisfies (H). Define

for t ∈ ℝ, x ∈ I. Then

Proof. (i) Fix an x ∈ I. Note that f-t(x), f+t(x)∈Ft(x) since the sets F^t(x) are closed. Hence, by Equation (1),

so φ(f±t(f±s(x)))=φ(x)+t+s=φ(f±t+s(x)). This implies that

for an α ∈ A or both belong to L^*, since I_α for α ∈ A are the intervals of constancy of φ. Obviously, in the second case, both values are equal. However, at the first case, f+t(f+s(x))≤sup Iα=f+t+s(x) and f-t+s(x)=inf Iα≤f-t(f-s(x)). On the other hand, putting f+s(x)=:y we have that f+t(y)∈Iα and f+t(y)∈Ft(y). Hence Ft(y)=Iα and f+t(y)=sup Iα≥f+t+s(x). This gives that

Similarly we prove that

(iv) Proving (i) we have shown that f±t(x) either belong to L^* or equals to one of the ends of the interval I_α which belong to L. Both cases give that f±t(x)∈L.

The remaining assertions are the simple consequences of formula (Equation 1).               □

Let φ be non-decreasing and non-injective surjection. Define the following family of functions

The index c_f is uniquely determined. This allows us to define

As a particular case of Proposition 2.2 in Farzadfard and Zdun [9] we get the following

Lemma 1

If f ∈ Realm(φ) then the following conditions are equivalent:

Let φ satisfy (H) and define

If T ≠ {0}, then T is a countable Abelian subgroup of group (ℝ, +).

In fact, since φ is constant in the intervals I_α, we have φ[I \ L*]={φ[Iα],α∈A}. It is easy to see that this set is unbounded above and below thus it is infinite and, consequently, countable since the intervals {I_α, α ∈ A} are pairwise disjoint.

Definition 1

A subgroup T given by Equation (5) is said to be a supporting group of the s-v iteration group {F^t : t ∈ ℝ}.

Theorem 3

Let T ≠ {0} be a supporting group of s-v iteration group {F^t : t ∈ ℝ} generated by a function φ satisfying (H). Then

Proof. (i) By Equation (2) f-t,f+t∈Realm(φ), indf±t=t for t ∈ ℝ and φ(f-t(x))=φ(f+t(x)). By Lemma 1 f±t(x)∈L* for x ∈ L^*. Since φ_|Iα is injective f-t(x)=f+t(x) for x ∈ L^*. Thus, by Proposition 3 (v), F^t(x) is a singleton belonging to L^*.

(ii) Let x ∈ I_α. By Lemma 1 f±t(x)∈Iβ for a β ∈ A. Thus Ft(x)⊂Iβ. If F^t(x) is a singleton then, by Proposition 1 (i), F^t(x) belongs to L^*, so f±t(x)∈L*, but this is a contradiction. Thus F^t(x) is a proper interval, so Ft(x)=Iβ.

(iii) Fix a β ∈ A. Since φ[I_β] is a singleton and φ is a surjection from I onto ℝ there exists an x ∈ I such that φ[I_β] = t+φ(x), that is Ft(x)=Iβ. Suppose x ∈ L^*. Then, by Lemma 1, f±t(x)∈L*, but this is a contradiction since f±t(x)∈Ft(x)=Iβ, so there exists an α ∈ A such that x ∈ I_α.

(iv) Since φ satisfies relation Equation (3) we have φ[L^*] = φ[F^t[L^*]] = φ[L^*] + t, so, by Lemma 1, t ∈ T.           □

Directly by Theorem 3 we get the following

Corollary 1

Let T ≠ {0} be the supporting group of the s-v group {F^t : t ∈ ℝ} with generating function satisfying (H). Then

Definition 2

A family of continuous mappings {f^t : I → I, t ∈ T} such that f^t ○ f^s = f^{t + s} for t, s ∈ T is said to be a T-iteration group.

Now we consider the problems connected with continuous selections of s-v iteration groups. The iteration groups {f-t,t∈ℝ} and {f+t,t∈ℝ} are the monotonic selections of s-v group {F^t, t ∈ ℝ} that is f±t(x)∈Ft(x), but they are discontinuous.

Let φ satisfies (H) and I_α = :[a_α, b_α] for α ∈ A be the intervals of constancy of φ. For t ∈ T define the affine mappings q_{t, α} : [a_α, b_α] → I such that

For every t ∈ T define the following mapping

Lemma 2

If T ≠ {0} is the supporting group of s-v group {F^t : t ∈ ℝ} generated by a function satisfying condition (H), then {q^t : I → I, t ∈ T} is a T-iteration group of continuous functions. Moreover, q^t(x) ∈ F^t(x) for t ∈ T and x ∈ I.

Proof. Note that qt,α[Iα]=Ft[Iα] and Ft[Iα1]<Ft[Iα2] if I_α1 < I_α2. Hence, by Theorem 3, it follows that the mappings q^t are strictly increasing surjections and, consequently, they are continuous.

It follows that for every t, s ∈ T, qt ○ qs[Iα]=qt[Fs[Iα]]=Ft[Fs[Iα]]=Ft+s[Iα]=qt+s[Iα]. Since the composition of affine functions is an affine function and there exists a unique increasing affine function mapping I_α onto the interval Ft+s[Iα] we get that q^t ○ q^s = q^{t + s} on I_α. Now it is easy to see that Proposition 3 implies our assertion.               □

Theorem 4

If s-v group {F^t : t ∈ ℝ} generated by a function satisfying condition (H) has a non trivial supporting group T, then there exists infinitely many disjoint T-iteration groups {f^t, t ∈ T} of continuous functions such that f^t(x) ∈ F^t(x) for t ∈ T and x ∈ I. T is a maximal additive group with this property.

Proof. Let γ : I → I be a homeomorphism such that γ(x) = x for x ∈ L and for every α ∈ A γ[I_α] = I_α. Put

It follows, by Lemma 2, that {f^t, t ∈ T} is a T-iteration group and f^t(x) ∈ F^t(x).

Let F^t have a continuous and strictly increasing selection f. Since for every α ∈ A, f[I_α] is a proper interval, Ft[Iα] is also an interval. Thus, by Corollary 1, t ∈ T.                         □

Let us make the following assumptions.

The conditions (i), (ii), and (iii) imply that φ is continuous and

Theorem 5

Let T be the supporting group of s-v group {F^t : t ∈ ℝ} generated by a function φ satisfying condition (H). If the group T is acyclic then the set L defined by (2) is a Cantor set and φ is a Lebesgue function which lives on L.

Proof. By Lemma 2 the family of mappings {q^t, t ∈ T} defined by Equation (6) is a disjoint T-iteration group. Denote by L_T the set of limit points of the orbits O(x) = {q^t(x) : t ∈ T}, i.e., LT=O(x)d. In Zdun [10] (see Th.1) it is proved that the set L_T does not depend on x and L_T is either a Cantor set in I or L_T = [a, b] or L_T = {a, b}. Moreover, L_T = {a, b} if and only if {q^t, t ∈ T} is a cyclic group (see [10] Theorem 2).

Since q^t(x) ∈ F^t(x) we have φ(q^t(x)) = φ(x) + t for x ∈ I. L_T ≠ [a, b]. In fact, suppose that L_T = [a, b]. Fix an x ∈ I and an interval I_α. By the density of the orbit O(x) there exist u, v ∈ ℝ such that u ≠ v and qu(x),qv(x)∈Iα. Hence φ(x) + u = φ(q^u(x)) = φ(q^v(x)) = φ(x) + v what is a contradiction.

By Proposition 1 (ii) and Lemma 2 the mapping Φ(t): = q^t is an isomorphism of T onto the group {q^t, t ∈ T}. Thus T is cyclic if and only if {q^t, t ∈ T} is cyclic, so T is cyclic if and only if L_T = {a, b}. Hence T is acyclic if and only if L_T is a Cantor set.

If T is acyclic then φ lives on L_T. Let x ∈ L_T and t ∈ T. Then qt(x)=f+t(x)∈L. Thus O(x) ⊂ L and, consequently, L_T ⊂ L, so L is also a Cantor set. By the assumption φ lives on L, however by the definition of q^t φ lives on L_F. Thus we get L_F = L.               □

Theorem 6

If f, g are commuting, iteratively incommensurable homeomorphisms and L_{f, g} ≠ cl I, then f and g are embeddable in a non-extensible disjoint T-iteration group {f^t, t ∈ T}, where T is a dense, countable subgroup of ℝ.

Proof. By Theorem 2 there exists an s-v iteration group {F^t : t ∈ ℝ} with continuous non-decreasing generating function φ such that f(x) ∈ F¹(x) and g(x) ∈ F^s(x) for an s ∉ ℚ and φ[L_{f, g}] = ℝ. Since L_{f, g} ≠ cl I, φ is a Lebesgue function which lives on L_{f, g}. Define T by Equation (5). By Theorem 5 f and g are embeddable in a T-iteration group {f^t, t ∈ T}. Since 1, s ∈ T the group T is dense.               □

In this note we consider the relation between the iteration groups of monotonic functions and the interval-valued iteration groups. These groups are still poorly investigated.

In Section 2 we indicate a desirability of the generalization of classical iteration groups in the real case. It is known that not all commutable iteratively incommensurable homeomorphisms are embeddable in an iteration group. However, Theorem 2 shows that the embeddabilty is always possible for s-v iteration groups.

Propositions 1 and 2 characterize s-v iteration groups of the form Equation (1). It is shown that, in our investigations, the form Equation (1) of s-v iteration groups are quite natural. Proposition 3 shows how s-v iteration groups of the form Equation (1) determine iterations groups of non-decreasing functions which are not injective.

A key concept of the paper is the supporting group T defined by Equation (5). If T is non-trivial additive group then it is countable and the set of all intervals of constancy of the generating function φ is also countable. Theorem 3 and Corollary 1 explain the meaning of the supporting group T. The restricted s-v group {F^t : t ∈ T} has a property that s-v functions F^t transform the intervals of constancy of the generating function φ onto itself and the points from its complement, that is the set L^*, onto singletons in L^*. Moreover, Theorem 4 and Corollary 1 show that each s-v function F^t for t ∈ T has continuous selection f^t such that family {f^t : t ∈ T} forms a group. Moreover, any F^t for t ∉ T has no continuous selection.

We have also proved that supporting group T is acyclic if and only if the generating function φ is a Lebesgue function which lives in a Cantor set.

The presented results may be helpful in the constructions of different iteration groups of non-decreasing functions.

MZ conceived the study and prepared the manuscript.