- Institute of Mathematics, Pedagogical University of Cracow, Kraków, Poland
Let I = (a, b) and L be a nowhere dense perfect set containing the ends of the interval I and let φ : I → ℝ be a non-increasing continuous surjection constant on the components of I \ L and the closures of these components be the maximal intervals of constancy of φ. The family {Ft, t ∈ ℝ} of the interval-valued functions Ft(x): = φ−1[t + φ(x)], x ∈ I forms a set-valued iteration group. We determine a maximal dense subgroup T ⊊ ℝ such that the set-valued subgroup {Ft, t ∈ T} has some regular properties. In particular, the mappings T ∍ t → Ft(x) for t ∈ T possess selections ft(x) ∈ Ft(x), which are disjoint group of continuous functions.
1. Introduction
A family of functions {ft : I → I, t ∈ ℝ} such that ft ○ fs = ft+s, t, s ∈ ℝ is said to be an iteration group, however a family of set-valued functions {Ft : I → 2I, t ∈ ℝ} such that Ft ○ Fs = Ft+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 {Ft : I → 2I, t ∈ ℝ} we find a special maximal additive subgroup T ⊂ ℝ such that group {Ft : I → 2I, t ∈ T} has several “regular” properties.
2. Materials and Methods
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 {Ft}.
THEOREM 1
The family {Ft : I → 2I} is a set-valued iteration group, i.e.,
where
Moreover, x ∉ Ft(x) for t ≠ 0.
Proof. Fix an x ∈ I. Let z ∈ Ft ○ Fs(x). Then there exists a y ∈ Fs(x) such that z ∈ Ft(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 ∈ Ft+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 Ft, t ∈ ℝ pairwise commute. This property is not transferible on the continuous selections of these set-valued mappings.
Let us assume that there exist Fu, Fv with which possess homeomorphic commuting selections f and g, that is f(x) ∈ Fu(x) and g(x) ∈ Fv(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 Lf, g does not depend on x and either this set is the interval cl I or Lf, 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 Lf, 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 {Ft, t ∈ ℝ} of type (1) such that f(x) ∈ F1(x) and g(x) ∈ Fs(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[Lf,g] = ℝ.
The proof is a simple consequence of Theorem 2 and Corollary 1 in Zdun [8].
The family {Ft, t ∈ ℝ} is a single-valued iteration group if and only if Lf, g = [a, b]. Then φ is strictly monotonic (see Zdun [8]).
In this paper we consider the case where Lf, g ≠ [a, b], that is {Ft : t ∈ ℝ} is a proper set-valued iteration group.
In the next section we will consider the more general case.
3. Results
Assume the following general hypothesis:
Then the function φ is continuous and the values of Ft are closed intervals or singletons. Denote by {Iα, α ∈ A} a family of the intervals of constancy of φ. These intervals are closed. Put
and
Note that is strictly increasing, φ[Iα] are singletons and if Iα < Iβ then φ[Iα] < φ[Iβ].
It is easy to verify that the s-v iteration group {Ft : I → cc[I], t ∈ ℝ} generated by φ has the following properties.
PROPOSITION 1
(i) For every x ∈ I Ft(x) either is a closed proper interval Iα or a singleton belonging to L*;
(ii) for every x ∈ I the s-v function t → Ft(x) is strictly decreasing, i.e., if s < t then for every u ∈ Fs(x) and v ∈ Ft(x), u < v;
(iii) for every x ∈ I
(iv) every s-v function Ft is constant on the intervals Iα;
(v) if s ≠ t then Ft(x) ∩ Fs(x) = Ø for x ∈ I, that is the group {Ft, t ∈ ℝ} is disjoint.
The conditions (i), (ii), (iii) characterize the interval-valued iteration groups. We have the following.
PROPOSITION 2
If an s-v iteration group {Ft, t ∈ ℝ} satisfies conditions (i), (ii), and (iii), where {Iα, α ∈ A} is a given family of closed disjoint proper intervals, then there exists a function φ satisfying (H) such that Ft are given by the formula (1).
Proof. Define
Let x0 ∈ I and put . Note that h is a bijection from ℝ onto . Define φ by the following way: if x ∈ Iα for an α ∈ A then , 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 Iα and
Since 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[Ft(x)] ⊂ φ[Ft(h(s)] = φ(x) + t, so
Since Ft(x) ⊂ φ−1[φ[Ft(x)]] we have Ft(x) ⊂ φ−1[φ(x) + t]. Note that φ−1[φ(x) + t] is a singleton or equals to one of the intervals Iα. If Ft(x) is a singleton then, by (i), for any α ∈ A. Thus φ−1[φ(x) + t] is not any of the intervals Iα, so it is a singleton. If Ft(x) is an interval Iα, then φ−1[φ(x + t)] must be also the same interval. This gives equality Ft(x) = φ−1[φ(x + t)]. □
PROPOSITION 3
Let a family of set-valued function Ft be given by (1), where φ satisfies (H). Define
for t ∈ ℝ, x ∈ I. Then
(i) the families and are iteration groups;
(ii) and for t ∈ ℝ are non-decreasing discontinuous functions constant on the intervals of constancy of φ;
(iii) the mappings are strictly decreasing;
(iv) , , t ∈ ℝ;
(v) , t ∈ ℝ.
Proof. (i) Fix an x ∈ I. Note that since the sets Ft(x) are closed. Hence, by Equation (1),
so . This implies that
for an α ∈ A or both belong to L*, since Iα for α ∈ A are the intervals of constancy of φ. Obviously, in the second case, both values are equal. However, at the first case, and . On the other hand, putting we have that and . Hence and . This gives that
Similarly we prove that
(iv) Proving (i) we have shown that either belong to L* or equals to one of the ends of the interval Iα which belong to L. Both cases give that .
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 cf 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:
(i) φ[L*] = φ[L*] + ind f;
(ii) φ[I \ L*] = φ[I \ L*] + ind f;
(iii) f[L*] = L*;
(iv) f maps each Iα into another one; moreover for every Iβ there exists Iα such that f[Iα] ⊂ Iβ.
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 Iα, we have . It is easy to see that this set is unbounded above and below thus it is infinite and, consequently, countable since the intervals {Iα, 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 {Ft : t ∈ ℝ}.
THEOREM 3
Let T ≠ {0} be a supporting group of s-v iteration group {Ft : t ∈ ℝ} generated by a function φ satisfying (H). Then
(i) if t ∈ T then for every x ∈ L* Ft(x) is a single point and Ft(x) ∈ L*;
(ii) if t ∈ T then for every α ∈ A there exists β ∈ A such that for x ∈ Iα;
(iii) if t ∈ T then for every β ∈ A there exists α ∈ A such that for x ∈ Iα;
(iv) if Ft[L*] = L* then t ∈ T.
Proof. (i) By Equation (2) , for t ∈ ℝ and . By Lemma 1 for x ∈ L*. Since φ|Iα is injective for x ∈ L*. Thus, by Proposition 3 (v), Ft(x) is a singleton belonging to L*.
(ii) Let x ∈ Iα. By Lemma 1 for a β ∈ A. Thus . If Ft(x) is a singleton then, by Proposition 1 (i), Ft(x) belongs to L*, so , but this is a contradiction. Thus Ft(x) is a proper interval, so .
(iii) Fix a β ∈ A. Since φ[Iβ] is a singleton and φ is a surjection from I onto ℝ there exists an x ∈ I such that φ[Iβ] = t+φ(x), that is . Suppose x ∈ L*. Then, by Lemma 1, , but this is a contradiction since , so there exists an α ∈ A such that x ∈ Iα.
(iv) Since φ satisfies relation Equation (3) we have φ[L*] = φ[Ft[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 {Ft : t ∈ ℝ} with generating function satisfying (H). Then
(i)
(ii)
(iii) T = {t ∈ ℝ:Ft[L*] = L*}.
DEFINITION 2
A family of continuous mappings {ft : I → I, t ∈ T} such that ft ○ fs = ft + 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 and are the monotonic selections of s-v group {Ft, t ∈ ℝ} that is , but they are discontinuous.
Let φ satisfies (H) and Iα = :[aα, bα] for α ∈ A be the intervals of constancy of φ. For t ∈ T define the affine mappings qt, α : [aα, bα] → I such that
For every t ∈ T define the following mapping
LEMMA 2
If T ≠ {0} is the supporting group of s-v group {Ft : t ∈ ℝ} generated by a function satisfying condition (H), then {qt : I → I, t ∈ T} is a T-iteration group of continuous functions. Moreover, qt(x) ∈ Ft(x) for t ∈ T and x ∈ I.
Proof. Note that and if Iα1 < Iα2. Hence, by Theorem 3, it follows that the mappings qt are strictly increasing surjections and, consequently, they are continuous.
It follows that for every t, s ∈ T, . Since the composition of affine functions is an affine function and there exists a unique increasing affine function mapping Iα onto the interval we get that qt ○ qs = qt + s on Iα. Now it is easy to see that Proposition 3 implies our assertion. □
THEOREM 4
If s-v group {Ft : 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 {ft, t ∈ T} of continuous functions such that ft(x) ∈ Ft(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[Iα] = Iα. Put
It follows, by Lemma 2, that {ft, t ∈ T} is a T-iteration group and ft(x) ∈ Ft(x).
Let Ft have a continuous and strictly increasing selection f. Since for every α ∈ A, f[Iα] is a proper interval, is also an interval. Thus, by Corollary 1, t ∈ T. □
Let us make the following assumptions.
(i) Let L be a Cantor set in I, that is L is a nowhere dense perfect set in I = (a, b) and a, b ∈ L.
(ii) Let Iω, ω ∈ ℚ be open pairwise disjoint intervals such that
(iii) Let φ:I → ℝ be a Lebesgue function which lives on a set L that is φ is a continuous non-increasing surjection constant on cl Iω and, let cl Iω be the maximal intervals of constancy of φ.
The conditions (i), (ii), and (iii) imply that φ is continuous and
THEOREM 5
Let T be the supporting group of s-v group {Ft : 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 {qt, t ∈ T} defined by Equation (6) is a disjoint T-iteration group. Denote by LT the set of limit points of the orbits O(x) = {qt(x) : t ∈ T}, i.e., . In Zdun [10] (see Th.1) it is proved that the set LT does not depend on x and LT is either a Cantor set in I or LT = [a, b] or LT = {a, b}. Moreover, LT = {a, b} if and only if {qt, t ∈ T} is a cyclic group (see [10] Theorem 2).
Since qt(x) ∈ Ft(x) we have φ(qt(x)) = φ(x) + t for x ∈ I. LT ≠ [a, b]. In fact, suppose that LT = [a, b]. Fix an x ∈ I and an interval Iα. By the density of the orbit O(x) there exist u, v ∈ ℝ such that u ≠ v and . Hence φ(x) + u = φ(qu(x)) = φ(qv(x)) = φ(x) + v what is a contradiction.
By Proposition 1 (ii) and Lemma 2 the mapping Φ(t): = qt is an isomorphism of T onto the group {qt, t ∈ T}. Thus T is cyclic if and only if {qt, t ∈ T} is cyclic, so T is cyclic if and only if LT = {a, b}. Hence T is acyclic if and only if LT is a Cantor set.
If T is acyclic then φ lives on LT. Let x ∈ LT and t ∈ T. Then . Thus O(x) ⊂ L and, consequently, LT ⊂ L, so L is also a Cantor set. By the assumption φ lives on L, however by the definition of qt φ lives on LF. Thus we get LF = L. □
THEOREM 6
If f, g are commuting, iteratively incommensurable homeomorphisms and Lf, g ≠ cl I, then f and g are embeddable in a non-extensible disjoint T-iteration group {ft, t ∈ T}, where T is a dense, countable subgroup of ℝ.
Proof. By Theorem 2 there exists an s-v iteration group {Ft : t ∈ ℝ} with continuous non-decreasing generating function φ such that f(x) ∈ F1(x) and g(x) ∈ Fs(x) for an s ∉ ℚ and φ[Lf, g] = ℝ. Since Lf, g ≠ cl I, φ is a Lebesgue function which lives on Lf, g. Define T by Equation (5). By Theorem 5 f and g are embeddable in a T-iteration group {ft, t ∈ T}. Since 1, s ∈ T the group T is dense. □
4. Discussion
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 {Ft : t ∈ T} has a property that s-v functions Ft 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 Ft for t ∈ T has continuous selection ft such that family {ft : t ∈ T} forms a group. Moreover, any Ft 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.
Author Contributions
MZ conceived the study and prepared the manuscript.
Conflict of Interest Statement
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
References
2. Olko J. On indexed functions generated by families of functions. Aequat Math. (2014) 88:125–35. doi: 10.1007/s00010-013-0239-1
3. Olko J. Selections of an iteration semigroup of linear set-valued functions. Aequat Math. (1998) 56:157–68. doi: 10.1007/s000100050052
4. Łydzińska G. On lower semicontinuity of some set-valued iteration semigroups. Nonlinear Anal. (2009) 71:5644–54. doi: 10.1016/j.na.2009.04.051
5. Piszczek M. On multivalued iteration semigroup. Aequat Math. (2011) 81:97–108. doi: 10.1007/s00010-010-0034-1
6. Zdun MC. On set-valued iteration groups generated by commuting functions. J Math Anal Appl. (2013) 398:638–48. doi: 10.1016/j.jmaa.2012.09.016
Keywords: iteration group, set-valued functions, simultaneous functional equations, Cantor set, singular Lebesgue function
Citation: Zdun MC (2016) On Singular Interval-Valued Iteration Groups. Front. Appl. Math. Stat. 2:13. doi: 10.3389/fams.2016.00013
Received: 24 May 2016; Accepted: 30 August 2016;
Published: 13 September 2016.
Edited by:
Witold Jarczyk, University of Zielona Góra, PolandReviewed by:
Ludwig Reich, University of Graz, AustriaKrzysztof Cieplinski, AGH University of Science and Technology, Poland
Copyright © 2016 Zdun. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Marek C. Zdun, bWN6ZHVuQHVwLmtyYWtvdy5wbA==