FUNCTIONS COMMUTING WITH AN ARBITRARY FIXED BIJECTION

Transcription

1 SARAJEVO JOURNAL OF MATHEMATICS Vol.9 (22) (203), DOI: /SJM FUNCTIONS COMMUTING WITH AN ARBITRARY FIXED BIJECTION ANDRZEJ MACH Abstract. In this note a characterization of all bijective functions commuting with a fixed bijection ϕ : X X, where X is an arbitrary nonempty set, is given. Introduction One can find commuting functions in many papers devoted to functional equations or iteration theory, see for example ], 2], 3], 5], 6]. Therefore, the problem of describing commuting functions with a given one seems to be interesting. The aim of this paper is a solution of the commutativity problem for all bijective functions. We present the form of commuting functions separately for closed and open orbits (see 4], page 5) of a given bijection ϕ : X X.. Functions commuting with a solution of Babbage functional equation.. Notations, definitions, lemma. Let us start with introducing some notations and definitions. Let X be an arbitrary nonempty set. Let ϕ : X X satisfy the Babbage equation (described in 4], page 288), this means (by ϕ n we denote n-th iteration). Let us define D := {m : m divides n} and ϕ n (x) = x, n 2, (.) X (m) := {x X : ϕ m (x) = x and ϕ k (x) x, for every k < m}. 200 Mathematics Subject Classification. Primary 39B2; Secondary 26A8. Key words and phrases. Bijective function, commuting function, iteration, closed and open orbits, Babbage functional equation.

2 222 ANDRZEJ MACH Lemma. (5] or 7]). The sets {X (m) : m D} are pairwise disjoint and the equality X (m) = X holds. m D For convenience of the reader we quote the proof from the paper 7]. Proof. It is evident that the sets {X (m) } m D are pairwise disjoint and m D X (m) X. To prove the reverse inclusion, we reason as follows: if x X and p {, 2,..., n} is the minimal number such that ϕ p (x) = x then, evidently, for t N {0} satisfying the inequalities we have Therefore n p p t n, p n tp p and ϕ tp (x) = x. ϕ n tp (x) = ϕ n tp (ϕ tp (x)) = ϕ n (x) = x. Hence n tp = p, whence p(t + ) = n. For every m D, by S m, we denote an arbitrary selection of the family of orbits { {x, ϕ(x), ϕ 2 (x),..., ϕ m (x)} : x X (m)}..2. Main constructions for a solution of the Babbage equation. Proposition.. Let us suppose that m D is fixed, X = X (m) and S = S m. Let g : S S be an arbitrary bijection. Moreover, let k : X {0,,..., m } be defined as follows k(x) := k ϕ k (x) S. (.2) The function g : X X defined by the formula g(x) := ϕ m k(x) g ϕ k(x)] (x) (.3) commutes with the function ϕ. Proof. Firstly, let us remark that (.2) implies k(ϕ(x)) = k(x) if k(x) 0 and k(ϕ(x)) = m if k(x) = 0. We have for k(x) 0 ( ) g ϕ(x) = ϕ m k(ϕ(x)) g ϕ k(ϕ(x))]( ) ϕ(x) = ϕ m k(ϕ(x)) g ϕ k(ϕ(x))+] (x) = ϕ m k(x)+ g ϕ k(x) +] ( (x) = ϕ ϕ m k(x) g ϕ k(x)] ) ( ) (x) = ϕ g(x).

3 FUNCTIONS COMMUTING WITH AN ARBITRARY FIXED BIJECTION 223 Similarly, for k(x) = 0 we have ( ) g ϕ(x) = ϕ m k(ϕ(x)) g ϕ k(ϕ(x))]( ) ϕ(x) = ϕ m k(ϕ(x)) g ϕ k(ϕ(x))+] (x) = ϕ m m+ g ϕ m +] ( (x) = ϕ ϕ m k(x) g ϕ k(x)] ) ( ) (x) = ϕ g(x). Remark.2. Let us remark that in the case m = we have ϕ = id X, S = X and by the formula (.3) we get g = g : X X, i.e. every bijection commutes with the identity function. Proposition.2. The function g given by the equality g = m D g m, where g m : X (m) X (m) has the form (.3) commutes with the function ϕ : X X such that eq. (.) holds. The proof results immediately from ϕ(x (m) ) = X (m), Lemma. and Proposition.. Example.3. Using Propositions. and.2 one can easily obtain the following: if g :] 2, ] 2, is an arbitrary bijection then the function g (x) for x > 2, 2 for x = 2, (.4) g ( x) for x < 2, commutes with the function ϕ(x) = x, for x R. functions x for x > 2, 2 for x = 2, 2 2 2x for x < 2, commute with the function ϕ(x) = x, for x R. Particularly, the 2 + ln(x + 2 ) for x > 2, 2 for x = 2, 2 ln( 3 2 x) for x < 2, Example.4. Let X = R \ {0} and ϕ(x) := x, x X. Let g : {, } {, } be an arbitrary bijection. Let us consider the following cases: a) S :=], 0 ]0,, or b) S :=], 0 ], +, or c) S :=], ]0,, or d) S :=], ], +. Let g2 : S S be an arbitrary bijection. Using Propositions. and.2 the function g (x) for x {, }, g2 (x) for x S, (.5) for x X \ S, g2 ( x )

4 224 ANDRZEJ MACH commutes with the function ϕ(x). Particularly, the functions x n, x, n for x =, for x =, sin π 2 x for x ], 0 ]0,, sin π for x ], ], +, 2x id {,} for x {, }, e x+ for x ],, ln x for x ]0,, for x ], 0, e x + for x ], +, ln x commute with the function ϕ(x) = x, for x R \ {0}. Example.5. Let A and Y be arbitrary nonempty sets and cardy 2. Let us write I p := {0,,..., p}, for p = 0,,.... Let X := A B, where B := {σ j i Y : i I 3, j I 4 } and cardb = 2. Let ϕ X X be defined by x, if x A, ϕ(x) := σ j (i+) 4, if x = σ j i, i I 3, j I 4 where (i + ) 4 denotes the remainder of division of i + by 4. Then ϕ 4 (x) = x, x X. In this case we have n = 4, X () = A, X (2) =, X (4) = B. Let g : A A be an arbitrary bijection. Let S := {σj 0 : j I 4}. Take the bijection g2 : S S defined as follows: g 2 (σj 0 ) := σ(j+) 5 0. According to Propositions. and.2 the function g : X X defined by the formula { g (x) for x A, commutes with ϕ. We define the relation in X by σ (j+) 5 (i) 4 for x = σ j i B, x y ϕ p (x) = y, for a p {0,,..., m }. One can easily observe that it is an equivalence relation. Let S be a selection of the family of orbits {{x, ϕ(x), ϕ 2 (x),..., ϕ m (x)} : x X}. Then there exists a unique function h S : X S such that h S (x) x. Note that h S (ϕ(x)) = h S (x). Theorem.6. Let X be an arbitrary nonempty set and n 2. Let ϕ : X X be a solution of the Babbage equation (.) and suppose that X =

5 FUNCTIONS COMMUTING WITH AN ARBITRARY FIXED BIJECTION 225 X (m) for a fixed m D. The bijection g : X X commutes with the solution ϕ if and only if there exist a selection S of the family of orbits {{x, ϕ(x), ϕ 2 (x),..., ϕ m (x)} : x X},a bijection g : S S and a function p : S {0,,..., m } such that g has the form g(x) := ϕ m k(x)+p(hs(x)) g ϕ k(x)] (x) (.6) where k : X {0,,..., m } is defined by (.2). Proof of the if part. Evidently, we have if x h S (x), h S (ϕ(x)) = h S (x) and further reasoning is similar as in Proposition.. We have for k(x) 0 ( ) g ϕ(x) = ϕ m k(ϕ(x))+p(hs(ϕ(x))) g ϕ k(ϕ(x))]( ) ϕ(x) = ϕ m k(ϕ(x))+p(hs(ϕ(x))) g ϕ k(ϕ(x))+] (x) = ϕ m k(x)++p(hs(x)) g ϕ k(x) +] (x) ( = ϕ ϕ m k(x)+p(hs(x)) g ϕ k(x)] ) ( ) (x) = ϕ g(x). Similarly, for k(x) = 0 we have ( ) g ϕ(x) = ϕ m k(ϕ(x))+p(hs(ϕ(x))) g ϕ k(ϕ(x))]( ) ϕ(x) = ϕ m k(ϕ(x))+p(hs(ϕ(x))) g ϕ k(ϕ(x))+] (x) = ϕ m m++p(hs(x)) g ϕ m +] (x) ( = ϕ ϕ m k(x)+p(hs(x)) g ϕ k(x)] ) ( ) (x) = ϕ g(x). Proof of the only if part. Let g : X X be a bijection commuting with ϕ. Take an arbitrary selection S of the family of orbits {{x, ϕ(x), ϕ 2 (x),..., ϕ m (x)} : x X}. One can observe that every function g : S S defined in such a way that g (s) = t for a t if and only if g(s) t, is a bijection. Let x X and put s := h S (x). There exists p(s) {0,,..., m } such that ϕ p(s) (g (s)) = g(s). Moreover we have ϕ k(x) (x) = s, ϕ m k(x) (s) = x. From the above ϕ m k(x)+p(s) g ϕ k(x)] (x) = ϕ m k(x)+p(s) g ( )] s = ϕ m k(x)( ) ( ) g(s) = g ϕ m k(x) (s) = ϕ m k(x)+p(s)( ) t = g(x).

6 226 ANDRZEJ MACH Theorem.7. Let X be an arbitrary nonempty set and n 2. Let ϕ : X X satisfy the Babbage equation (.). The bijection g : X X commutes with the solution ϕ if and only if g = m D g m, where g m : X (m) X (m) have the form (.6). The proof results immediately from ϕ(x (m) ) = X (m), Lemma. and Theorem.6. Example.8. Let X = B, where B is defined as in Example.5. Let ϕ be defined by ϕ(σ j i ) = σj (i+) 4, for i I 3, j I 4. Let S be the same as in Example.5, so S := {σ j 0 : j I 4}. Take the bijection g : S S defined as follows: g (σ j 0 ) := σ(j+2) 5 0. Define the function p : S {0,, 2, 3} by p(σ j 0 ) = (j) 4. By the formula (.6) we get the function g(σ j i ) = σ(j+2) 5 (j+i) 4. One can easily verify that g commutes with ϕ. 2. The case of open orbits Let X be an arbitrary nonempty set. Let ϕ : X X be a bijection such that ϕ n (x) x, n. (2.) In this case all orbits are as follows: {..., ϕ 3 (x), ϕ 2 (x), ϕ (x), x, ϕ(x), ϕ 2 (x), ϕ 3 (x), ϕ 4 (x),... }. Now, we define the equivalence relation in X by x y ϕ p (x) = y, for a p Z. Let S be a selection of the family of orbits { } {..., ϕ 3 (x), ϕ 2 (x), ϕ (x), x, ϕ(x), ϕ 2 (x), ϕ 3 (x), ϕ 4 (x),... } : x X. Then there exists a unique function h S : X S such that h S (x) x. Note that h S (ϕ(x)) = h S (x). Theorem 2.. Let X be an arbitrary nonempty set. Let ϕ : X X be a bijective function satisfying (2.). The bijection g : X X commutes with the function ϕ if and only if there exist a selection S of the family of orbits { {..., ϕ 3 (x), ϕ 2 (x), ϕ (x), x, ϕ(x), ϕ 2 (x), ϕ 3 (x), ϕ 4 (x),... } : x X a bijection g : S S and a function p : S Z such that g has the form g(x) := ϕ k(x)+p(hs(x)) g ϕ k(x)] (x) (2.2) where k : X Z is defined by (.2). },

7 FUNCTIONS COMMUTING WITH AN ARBITRARY FIXED BIJECTION 227 Proof of the if part. Evidently, we have if x h S (x), h S (ϕ(x)) = h S (x) and further reasoning is similar as in Proposition. and in Theorem.6. Indeed, we have k(ϕ(x)) = k(x) for all x X, so ( ) g ϕ(x) = ϕ k(ϕ(x))+p(hs(ϕ(x))) g ϕ k(ϕ(x))]( ) ϕ(x) = ϕ k(ϕ(x))+p(hs(ϕ(x))) g ϕ k(ϕ(x))+] (x) = ϕ k(x)++p(hs(x)) g ϕ k(x) +] (x) ( = ϕ ϕ k(x)+p(hs(x)) g ϕ k(x)] ) ( ) (x) = ϕ g(x). Proof of the only if part. Let g : X X be a bijection commuting with ϕ. Take an arbitrary selection S of the family of orbits { } {..., ϕ 3 (x), ϕ 2 (x), ϕ (x), x, ϕ(x), ϕ 2 (x), ϕ 3 (x), ϕ 4 (x),... } : x X. One can observe that every function g : S S defined in such a way that g (s) = t for a t if and only if g(s) t, is a bijection. Let x X and put s := h S (x). There exists p(s) Z such that ϕ p(s) (g (s)) = g(s). Moreover we have ϕ k(x) (x) = s, ϕ k(x) (s) = x. From the above ϕ k(x)+p(s) g ϕ k(x)] (x) = ϕ k(x)+p(s) g ( )] s = ϕ k(x)+p(s)( ) t = ϕ k(x)( ) ( ) g(s) = g ϕ k(x) (s) = g(x). 3. Final remarks, examples and problems Remark 3.. The results of this paper can be useful in finding all functions commuting with a given one. Namely, by Theorem.7, we have a ready to use description of commuting functions for all closed orbits of the given bijective function. Similarly, in Theorem 2., we have a ready to use construction of commuting functions for all open orbits of the given bijective function. Since the union of the closed orbits X I and the union of open orbits X II are disjoint and ϕ(x I ) = X I, ϕ(x II ) = X II, then the presented results are complete for the characterization of all commutative functions

8 228 ANDRZEJ MACH with a given bijection ϕ. More precisely, the domain X of the bijection ϕ can be decomposed onto disjoint sets as follows X = X (n) X ( ), n= where X (n) = {x X : ϕ n (x) = x and ϕ k (x) x for 0 < k < n} and X ( ) = {x X : ϕ n (x) x for n }. Theorems.6,.7 and Remark.2 describe the form of commuting function g on the sets X (n), n and Theorem 2. describes the form of g on the set X ( ). Example 3.2. Let us consider the bijection ϕ : R + R + given by the formula ϕ(x) = 3x. Using the presented theorems we can construct all functions g commuting with ϕ. We have R + = R () + R( ) +, where R() + = {0} and R ( ) + = R + \{0}. Therefore g(0) = 0 and to define g on the set R ( ) + we use Theorem 2.. The interval, 3 forms a selection of the family of orbits { {..., ϕ 3 (x), ϕ 2 (x), ϕ (x), x, ϕ(x), ϕ 2 (x), ϕ 3 (x), ϕ 4 (x),... } : x R ( ) + Moreover R ( ) + = l Z3 l, 3 l+ and for x 3 l, 3 l+, l Z - according with (.2) - we have k(x) = l. The formula below gives a family of functions which commute with ϕ. { 0 for x = 0, 3 p(s)+l g (3 l x) for x 3 l, 3 l+ and l Z, where g :, 3, 3 is an arbitrary bijection and p :, 3 Z is an arbitrary function. Taking the bijection g :, 3, 3 defined by the formula g (x) = + 2 sin π 4 (x ) and taking the function p(s) = 7, for s, 3, we obtain from the above the form of the commuting function g as follows { 0 for x = 0, 3 7+l + 2 sin π 4 (3 l x )] for x 3 l, 3 l+ and l Z. Example 3.3. For the bijection ϕ : R + R + given by the formula ϕ(x) = x 3 we have R + = R () + R( ) +, where R() + = {0, } and R( ) + = R + \ {0, }. Let g : {0, } {0, } be an arbitrary bijection. Let g 2 : 8, 2 2, 8 8, 2 2, 8 be an arbitrary bijection and p : 8, 2 2, 8 Z be an arbitrary function. We have R ( ) + = 2 3l+, 2 3l l Z l Z2 3l, 2 3l+ }.

9 FUNCTIONS COMMUTING WITH AN ARBITRARY FIXED BIJECTION 229 and k(x) = l, for x 2 3l+, 2 3l 2 3l, 2 3l+, l Z. The formula below gives a family of functions which commute with ϕ. { g (x) for x {0, }, g2 (x3 l )] 3p(s)+l for x 2 3l+, 2 3l 2 3l, 2 3l+ and l Z. Example 3.4. For the bijection ϕ : R R given by the formula ϕ(x) = x 3 we have R = R () R (2) R ( ), where R () = {0}, R (2) = {, } and R ( ) = R \ {, 0, }. Theorems.6,.7 and Remark.2 describe the form of the commuting function g on the sets R (), R (2) and Theorem 2. describes the form of g on the set R ( ). Determining the parameters as in the previous examples we can obtain all functions commuting with the given function ϕ. Problem 3.5. Characterize all functions commuting with a given finction f : X X (not necessarily bijective), where X is an arbitrary nonempty set, particularly for f : R R. Acknowledgements. I wish to thank to anonymous referee for all valuable remarks. References ] D. Krassowska and M. C. Zdun, On limit sets of mixed iterates of commuting mappings, Aequationes Math., 78 (2009), ] D. Krassowska and M. C. Zdun, On the embeddability of commuting continuous injections in iteration semigroups, Publicationes Math., 75 (2009), ] D. Krassowska, Iteration groups and commuting functions in R n, Grazer Math. Berichte, 35 (2007), ] M. Kuczma, Functional Equations in a Single Variable, Monografie Mat. 46, Polish Scientific Publishers (PWN), Warszawa ] A. Mach, The translation equation on certain n-groups, Aequationes Math., 47 (994), 30. 6] A. Mach, The construction of the solutions of the generalized translation equation, Aequationes Math., 64 (2002), 23. 7] A. Mach, On some functional equations involving Babbage equation, Result. Math., 5 (2007), (Received: November 5, 202) (Revised: February 20, 203) Faculty of Economics Department of Social and Technical Science University of Business and Enterprise ul. Akademicka Ostrowiec Świetokrzyski Poland amach@wsbip.edu.pl or Faculty of Engineering and Economics

10 230 ANDRZEJ MACH The State Higher School of Vocational Education Narutowicza Ciechanów Poland andrzej.mach@pwszciechanow.edu.pl