Iteration of Functions f: X k →X and their Periodicity
IIteration of Functions f : X k → X and their Periodicity Suneil Parimoo ∗ This version: October 25, 2020
Abstract
We propose a notion of iterating functions f : X k → X in a way that represents recurrencerelations of the form a n + k = f ( a n , a n +1 , ..., a n + k − ) . We define a function as n -involutory when its n th iterate is the identity map, and discuss elementary group-theoretic properties of such functionsalong with their relation to cycles of their corresponding recurrence relations. Further, it is shownthat a function f : X k → X that is -involutory in each of its k arguments (holding others fixed)is ( k + 1) -involutory. Mathematics Subject Classification (2020):
Primary: 39B12; Secondary: 05E05, 20A05,26A18, 30D05, 37B20, 37C25
Keywords:
Iterative functional equations, iterative roots, involution, periodic function, symmet-ric function, Babbage equation, recurrence relation, difference equation, fixed point, cycle Introduction
This paper proposes a means to extend the notion of function iteration to functions f : X k → X forsome set X and integer k ≥ , and aims to explore such functions that obey a certain iterative periodicproperty.Function iteration is well defined when functions are self-maps. For a self-map f : X → X , the n thiterate of f , denoted by f n for some nonnegative integer n , is defined recursively by f ≡ id X and f n +1 ≡ f ◦ f n , ∗ Columbia University. Email: [email protected] a r X i v : . [ m a t h . G M ] O c t here id X is the identity map on X . If f is invertible with inverse f − , then this definition extends tonegative iterates, where the − n th iterate of f is the n th iterate of f − . The associativity of functioncomposition immediately gives the following properties for integers m and n : . Addition rule: f m ◦ f n = f n ◦ f m = f m + n (1) . Multiplication rule: ( f m ) n = ( f m ) n = f mn . (2)The multiplication rule (2) provides a natural means of extension to fractional iterates; for integers n (cid:54) = 0 and m , with gcd ( m, n ) = 1 , an mn th iterate of f is any function g such that g n = f m (c.f. Isaacs(1950)).Function iteration for self-maps may be understood as a representation of a recurrence relation.For a sequence { a n } n ∈ N defined recursively through some self-map f by a n +1 = f ( a n ) given some seed value a , the n th term of the sequence may be computed as a n = f n ( a ) . In extending the notion of function iteration to a function f : X k → X , it is desirable for the iteratesto likewise represent the state of a recurrence relation. One way to define its iterates is by defining itsfirst iterate, f , as the self-map on X k given by f : ( x , x , ..., x k ) → ( f ( x , x , ..., x k ) , ..., f ( x , x , ..., x k (cid:124) (cid:123)(cid:122) (cid:125) k times ) , (3)and defining other iterates of f as typical iterates of f . Iterates thus defined treat the k arguments in asymmetric manner and consequently feature redundancies in that each of their k component functionsare identical. However, it is not immediately clear what distinct application this definition of functioniteration serves. Instead, the definition of function iteration for a function f : X k → X that is offeredin this paper has the natural interpretation of representing k th order recurrence relations of the form a n + k = f ( a n , a n +1 , ..., a n + k − ) . (4) Defining function iteration as in (3) can be understood as representing a recurrence relation a n +1 = f ( a n , ..., a n (cid:124) (cid:123)(cid:122) (cid:125) k times ) .However, such a system can be written succinctly as a n +1 = g ( a n ) for self-map g and thus can be represented by usualfunction iteration. f as a self-map on X k that produces k consecutive terms ofthe recurrence relation, and defining iterates of f as typical iterates of f . A formal definition withexamples is presented in section 2.A self-map f : X → X is said to be involutory (or is an involution ) if it is its own inverse: f = id X (Aczel (1948)). The immediate generalization of this property is that a self-map’s ( n − )th iterate isidentical to its own inverse for positive integer n : f n = id X . (5)Treating f as an unknown, (5) defines a functional equation known as Babbage’s functional equation (Babbage (1815), Babbage (1816), Babbage (1820), Babbage and Gergonne (1822)) and its solutionshave been well studied in the literature. Łojasiewicz (1951) provides the general construction of thesolution (see also Bogdanov (1961), and for certain real solutions, see the earlier work Ritt (1916)).Much of the work of the twentieth century relating to Babbage’s equation—and more broadly, thetheory of functional equations—can be found in the monograph Kuczma et al. (1990) (c.f. also Baronand Jarczyk (2001) for a further development of this work).A solution f of (5) is generally referred to as an nth iterative root of identity (Kuczma et al. (1990)),and when n is the smallest positive integer such that f satisfies (5), f is variously known as a functionof order n or said to circulate with period n (Ritt (1916)), or is said to be periodic with period n (McShane (1961)). In this paper, we introduce and mostly use the terminology involutory of order n ,or n -involutory. As defined in section 3, a function f : X k → X is said to be n -involutory when its n thiterate is the identity map. That is, f is n -involutory when f , a self-map on X k , satisfies Babbage’sequation throughout its domain, X k . More generally, a function f : X k → X is n -involutory at point x when f satisfies Babbage’s equation at the specified point x ∈ X k . In section 3.1, I discuss elementary group-theoretic properties of self-maps that are n -involutory.In this context, I also provide a concise proof that continuous self-maps on R that are involutory ofan integral order must in fact be involutory. This latter result, Proposition 1, is well-known in theliterature on functional equations (c.f. Ewing and Utz (1953), Vincze (1959), McShane (1961)), andis stated in Theorem 11.7.1 in Kuczma et al. (1990). Proposition 1 supplies general motivation inthe sequel for approaching the question of when multivariate functions f : X k → X have a commoninvolutory order (only possibly depending on k ). In section 3.2, I consider elementary properties of This terminology may seem redundant, although there are some subtle distinctions between being n -involutory andbeing periodic with period n . To begin with, I apply the term “ n -involutory” to describe f , which need not be a self-map(if k > ), since the focus of the paper is on a sufficient condition on f under which the self-map f satisfies Babbage’sequation. I also do not restrict the order n to be minimal in stating that a function is n -involutory. More conceptually,however, the term “ n -involutory” reminds the reader that the property it signifies is ultimately a generalization of beinginvolutory, and further, as will be made clear in Proposition 2, that involutory self-maps on X directly give rise tofunctions f : X k → X having this property. f : X k → X that are n -involutory. In the spirit of Proposition 1, the main result obtainedin this section is Proposition 2, which asserts that when multivariate functions are involutory in eachof their k arguments (a property I refer to as being induced involutory , defined more precisely in thesequel), they are involutory of common order k +1 . Section 3.3 discusses the relation between functionsthat are n -involutory at a point and cycles of the recurrence relations that they represent. Section 4concludes with discussion on possible extensions and some research questions. Definition of iteration of functions f : X k → X Consider the recurrence relation in (4). Observe that the state of the system is characterized bya k -tuple of the elements of the sequence itself. Our definition of function iteration for functions f : X k → X is based on representing such systems and is given as follows: Definition 1.
For a set X and function f : X k → X , the n th iterate of f , denoted by f n fornonnegative integer n , is defined as the n th iterate of the self-map f : X k → X k , given by f : ( x , x , ..., x k ) → ( f , f , ..., f k ) ,f = f ( x , x , ..., x k ) ,f = f ( x , x , ..., x k , f ) , ... f j = f ( x j , x j +1 , ..., x k , f , f , ..., f j − ) ... f k = f ( x k , f , f , ..., f k − ) . Note that while f itself is not a self-map, its iterates thus defined are self-maps and satisfy theaddition and multiplication rules, (1) and (2). This definition hence also allows for an immediateextension to negative and fractional iterates in the way described before. In the event the function f is a self-map ( k = 1 ), this definition naturally concurs with the usual definition of function iteration.The graphical construction of f for f : R → R is illustrated in Figure (1).Definition (1) allows us to represent recurrence relations described by (4) as an iterated function: ( a nk +1 , a nk +2 , ..., a nk + k ) = f n ( a , a , ..., a k ) . (6)4igure 1: Zeroth and first iterates (underlined in blue) at point ( a, b ) ∈ R of function (red) f : R → R Example 1.
Consider the function f : C → C given by f ( x , x ) = x + x . Its iterates encode terms of the recurrence relation a n +2 = a n + a n +1 , and are given by f n ( x , x ) = ( F n − x + F n x , F n x + F n +1 x ) , where F n is the n th Fibonacci number, where F = 0 and F = 1 . Example 2.
Consider the function f : X k → X for some set X given by f ( x , x , ..., x k ) = g ( x j ) , j ∈ { , , ..., k } for some function g : X → X . The iterates of f encode terms of the recurrence relation a n + k = g ( a n + j − ) . In the case j = 1 , the iterates are given by f n ( x , x , ..., x k ) = ( g n ( x ) , g n ( x ) , .., g n ( x k )) , j = k , the iterates (for positive integer n ) are given by f n ( x , x , ..., x k ) = ( g nk − k +1 ( x k ) , g nk − k +2 ( x k ) , .., g nk ( x k )) . Example 3.
Consider the function f : C k → C given by f ( x , x , ..., x k ) = A − k (cid:88) j =1 x j , for some constant A ∈ C , whose iterates encode terms of the recurrence relation a n + k = A − (cid:80) k − i =0 a n + i .Its iterates are cyclical, such that for any integer m , they are given by f m ( k +1) ( x , x , ..., x k ) = ( x , x , ..., x k ) f m ( k +1)+1 ( x , x , ..., x k ) = ( A − k (cid:88) j =1 x j , x , x , ..., x k − ) ... f m ( k +1)+ p ( x , x , ..., x k ) = ( x k − p +2 , x k − p +3 , ..., x k , A − k (cid:88) j =1 x j , x , x , ..., x k − p ) ... f m ( k +1)+ k ( x , x , ..., x k ) = ( x , x , ..., x k , A − k (cid:88) i =1 x i ) . Cycles such as those in the third example supply the motivation for exploring functions f : X k → X featuring such iterative periodicity, as pursued in section 3. Periodicity3.1 Periodicity of self-maps
Based on the notion of function iteration established in Definition (1), I offer the following definitionas a generalization of involutory functions:
Definition 2.
A function f : X k → X is involutory of order n , or n -involutory , for integer n when f n = id X k . The integer n is referred to as an involutory order of f .A function that is n -involutory is thus one whose ( n − th iterate is the inverse map of its firstiterate. More generally, such a function can be characterized as one whose j th iterate is the inverse map6f its ( n − j ) th iterate for any integer j . Note that our definition is based on integral involutory ordersto ensure well-definedness. Of course, every function is -involutory. If f has a positive involutoryorder, then when n is the smallest positive involutory order of f (equivalently, when f is periodicwith period n as per McShane (1961)), the set { id X k , f , f , ..., f n − } , endowed with the composition operation, ◦ , comprises a cyclical group of order n generated by f .The following are immediate properties of such functions when they are self-maps: Lemma 1.
Let f : X → X be n -involutory for n ∈ Z . Then we have the following:1. f is nm -involutory for any m ∈ Z .2. For m ∈ Z , f m is ngcd ( n,m ) -involutory.3. The identity map, id X , is m -involutory for any m ∈ Z .4. If f is also m -involutory for m ∈ Z , and gcd ( m, n ) = 1 , then f = id X .5. For any invertible function g : Y → X , the map g − ◦ f ◦ g is n -involutory. The properties of Lemma 1 are apparent from the aforementioned group structure. The fifthproperty is a conjugacy property that has been noted as early as Babbage’s work; in particular, when Y = X , it reveals that any self-map on set X that is n -involutory gives rise to a conjugacy class of otherfunctions that are also n -involutory within the symmetric group on X . For example, the involutoryself-map h defined on X ≡ R > , given by h : x → ln (cid:16) e x +1 e x − (cid:17) , is conjugate to the linear self-map − id X since h = g − ◦ ( − id X ) ◦ g , where g = g ◦ g ◦ g ◦ g , with g : x → x − , g : x → log ( x ) , g : x → x − , g : x → e x . In fact, it turns out all strictly decreasing involutions on a real interval are conjugateto the negative identity map and hence are conjugate to each other (c.f. Theorem 11.7.3 in Kuczmaet al. (1990)). Another result, perhaps more surprising, is that all complex rational self-maps that are n -involutory fall into one of three simple explicit conjugacy classes (c.f. Theorem 11.7.4 in Kuczmaet al. (1990)). Exploring this conjugacy property, and particularly determining when a function is linearizable (or conjugate to a linear map), has given rise to a fruitful area of research within thetheory of functional equations (see, for instance, the recent work Homs-Dones (2020), which gives areview of several main results in this area for periodic functions, and looks at linearization of functionssatisfying a generalization of Babbage’s equation (5)).While Lemma 1 applies to functions that are involutory of arbitrary order n , for continuous self-maps on R , a function that is involutory of some integral order must be involutory of order 2 (i.e.involutory), as the next proposition establishes. 7 roposition 1. Let f : I → I be a continuous self map on interval I ⊂ R . Then f is n -involutory forsome integer n if and only if either 1. f = id I or 2. n is even and f is a strictly decreasing involution.Proof. ( ⇐ =) Sufficiency is trivial by properties 1 and 3 of Lemma 1. ( = ⇒ ) Necessity is established in steps as follows:Step 1: f is strictly monotoneBy continuity, it suffices to note that f is injective, which follows from the invertibility of f .For the remaining steps, assume that f (cid:54) = id I i.e. ∃ z ∈ I s.t. f ( z ) ≷ z ; we will show n is even and f is a strictly decreasing involution.Step 2: f is strictly decreasingAssume by contradiction that f is strictly increasing. Then f ( z ) ≷ z = ⇒ f ( z ) ≷ f ( z ) ≷ z = ⇒ ... = ⇒ z = f n ( z ) ≷ f n − ( z ) ≷ ... ≷ f ( z ) ≷ z , a contradiction.Step 3: n is evenAssume by contradiction n is odd. By repeated application of step 2, we have f ( z ) ≷ z = ⇒ z = f n ( z ) ≷ f n − ( z ) ≶ ... ≶ f ( z ) ≷ z = f n ( z ) . The first inequality in this chain thus asserts that z ≷ f n − ( z ) , while the last inequality in this chain asserts f ( z ) ≷ f (cid:0) f n − ( z ) (cid:1) , contradicting that f is strictly decreasing.Step 4: f = id I Assume by contradiction that ∃ z (cid:48) ∈ I s.t. f ( z (cid:48) ) ≷ f n ( z (cid:48) ) = z (cid:48) . By step 2, applying f − to bothsides of this inequality implies f ( z (cid:48) ) ≶ f n − ( z (cid:48) ) . Applying the inverse again yields f n ( z (cid:48) ) = z (cid:48) ≷ f n − ( z (cid:48) ) . Repeatedly applying the inverse and noting n is even (step 3) implies f ( z (cid:48) ) ≷ f n ( z (cid:48) ) ≷ f n − ( z (cid:48) ) ≷ ... ≷ f ( z (cid:48) ) , a contradiction.As mentioned in section 1, the result given in Proposition 1 is well established in the literature.Presenting the result as such supplies some motivation in the sequel, where, in the general spirit ofthis proposition, I obtain a sufficient condition under which functions f : X k → X are involutory of acommon order. f : X k → X In this section, we consider properties of functions f : X k → X that are n -involutory. A function f being n -involutory is equivalent to the self-map f being n -involutory, so the properties of Lemma 1extend quite naturally to such functions, as stated in the following lemma: Lemma 2.
Let f : X k → X be n -involutory for n ∈ Z . Then we have the following:1. Properties 1-2 of Lemma 1 hold. . The map ˆ id X k : X k → X , defined as ˆ id X : ( x , x , ..., x k ) → x , is m -involutory for any m ∈ Z .3. If f is also m -involutory for m ∈ Z , and gcd ( m, n ) = 1 , then f = ˆ id X k .4. For ˜ f : X → X any involutory function, the map ˜ f ◦ ˆ id X k is -involutory.5. For g : Y → X any invertible function, let ˜ g : Y k → X k be given by ˜ g : ( y , y , ..., y k ) → ( g ( y ) , g ( y ) , ..., g ( y k )) . Then the map g − ◦ f ◦ ˜ g : Y k → Y is n -involutory. It is worth noting that properties 2-4 reveal that ˆ id X k acts as a kind of identity map for functions f : X k → X , since ˆ id X k has its first iterate given by id X k . However, a map ( x , x , ..., x k ) → x j for j (cid:54) =1 generally need not be involutory of an integral order. This contrast results from the asymmetricmanner in which the iterates of f treat their arguments. While the properties of Lemma 1 thus extendto functions f : X k → X , Proposition 1 does not immediately extend in that continuous functionsin Euclidean space that are involutory of an integral order need not be involutory of some commonorder (depending only possibly on k ). For example, consider the function f : R k → R given by f : ( x , x , ..., x k ) → A − x for some constant A ∈ R , which is -involutory by property 4 of Lemma 2.In contrast, the function f : R k → R given by f : ( x , x , ..., x k ) → A − (cid:80) ki =1 x i is ( k + 1) -involutory,as per Example 3. However, Example 3 displays certain properties that suggest sufficient conditionsunder which one may obtain a notion of a common involutory order, in the spirit of Proposition 1. Wedefine a few terms to facilitate understanding these properties: Definition 3.
Given function f : X k → X , let f j ( ·| x − j ) : X → X denote the induced function f j ( x j | x − j ) ≡ f ( x , x , ..., x k ) where j ∈ { , , ..., k } and x − j ≡ { x , x , ..., x k }\{ x j } ∈ X k − is fixed. The function f : X k → X is induced involutory of order n in argument j (written II- n { j }) when the induced function f j ( ·| x − j ) is n -involutory for any x − j ∈ X k − . The function is induced involutory of order n (written II- n ) when itis II- n { j } for every j ∈ { , , ..., k } . The function is induced involutory in argument j (written II-{ j })when it is II- { j } . The function is induced involutory (II) when it is II- . In other words, a function is II- n { j } when it is n -involutory in argument j , holding fixed theother arguments. Also, recall that a function f : X k → X is said to be symmetric when its valueis unchanged by any permutation of its k arguments. The following lemma establishes the relationbetween II- n functions and symmetry. 9 emma 3. If f : X k → X is symmetric and II- n { j } for integer n and j ∈ { , , ..., k } , then f is II- n .If f : X k → X is II, then it is symmetric.Proof. The first claim is immediate since symmetry permits the arguments of f to be transposed sothat for any x , x , ..., x k ∈ X and any j (cid:48) ∈ { , , ..., k } , we have x j = f nj ( x j | x − j ) = f nj (cid:48) ( x j | x − j ) .It suffices to show the second claim for k = 2 since any permutation of k > arguments is acomposition of pairwise transpositions. Note that f being II implies f is invertible with respect toeach argument. Observe that for any x , x ∈ X , we have x = f ( f ( x , f ( x , x )) , f ( x , x )) since f is II-{ }, while we also have x = f ( x , f ( x , x )) since f is II-{ }. Equating the two expressions for x and applying the inverse of f ( · , f ( x , x )) toboth implies x = f ( x , f ( x , x )) . We also have x = f ( x , f ( x , x )) since f is II-{ }. Equating the two expressions for x and applying the inverse of f ( x , · ) to bothimplies f ( x , x ) = f ( x , x ) .Note that the second claim in Lemma 3 does not apply to II- n functions for integers n ≥ . Considerthe following example: Example 4.
Suppose f : C → C is given by f : ( x , x ) → ax + bx , where a and b are distinct n th roots of unity for integer n ≥ , neither of which is unity itself. Then the n th iterates of f withrespect to each argument are given as f n ( x | x ) = a n x + b − a n − a x = x ,f n ( x | x ) = b n x + a − b n − b x = x , so that f is II- n but not symmetric.The following proposition asserts that II functions are involutory of a common order. Proposition 2. If f : X k → X is II, then f is ( k + 1) -involutory. In fact, k + 1 is the minimal (positive integral) involutory order of f , so that f is periodic with period k + 1 . roof. Let x denote ( x , x , ..., x k ) ∈ X k , and let x − j denote { x , x , ..., x k }\{ x j } ∈ X k − . By Lemma3, f is symmetric, so for any j ∈ { , , ..., k } , we have f ( f ( x ) , x − j ) = x j (independent of the order ofarguments). Consequently, the first k + 1 iterates are computed as follows: f ( x , x , ..., x k ) = ( f ( x ) , x , x , ..., x k − ) ... f p ( x , x , ..., x k ) = ( x k − p +2 , x k − p +3 , ..., x k , f ( x ) , x , x , ..., x k − p ) ... f k ( x , x , ..., x k ) = ( x , x , ..., x k , f ( x )) f k +1 ( x , x , ..., x k ) = ( x , x , ..., x k ) Proposition 2 thus permits a way to generate functions that are involutory of any order k + 1 bydefining a function f : X k → X that is a direct analogue of an involutory self-map on X , as in Example3, which is the analogue of the self-map f : x → A − x. One can see how Proposition 2 breaks downwhen the II assumption is violated. Consider the following examples:
Example 5.
Let f : C → C be given as f : ( x , x ) → ax + bx , where a = − i √ and b = − − i √ .By Example 4, f is II-3, but it is not involutory of any integral order, since its iterates are given by f n ( x , x ) = ( F n − a n x + F n a n +1 x , F n a n +2 x + F n +1 a n x ) , where F n is the n th Fibonacci number, with F = 0 and F = 1 . Example 6.
Consider the symmetric function f : X → X specified below, where X ≡ { x , x , x } . f x x x x x x x x x x x x x x x The value of f ( x i , x j ) for ( x i , x j ) ∈ X is read as the table element corresponding to row x i andcolumn x j . It is easily verified that f is II-3 since fixing any row or column, the elements form a3-cycle. For instance, the 3-cycle corresponding to the third row is x → x → x → x . Moreover,11 is -involutory since all the elements of the set X endowed with the self-map f are generated aspart of two 4-cycles and one singleton cycle: ( x , x ) → f : ( x , x ) → f : ( x , x ) → f : ( x , x ) → f : ( x , x );( x , x ) → f : ( x , x ) → f : ( x , x ) → f : ( x , x ) → f : ( x , x );( x , x ) → f : ( x , x ) . Example 7.
Consider the function f : X → X specified below, where X ≡ { x , x , x , x } . f x x x x x x x x x x x x x x x x x x x x x x x x Here, f is both symmetric and persymmetric (i.e. symmetric along the antidiagonal) and II-3.Moreover, f is -involutory since all the elements of the set X endowed with the self-map f aregenerated as part of a 15-cycle and singleton cycle: ( x , x ) → f : ( x , x ) → f : ( x , x ) → f : ( x , x ) → f : ( x , x ) → f : ( x , x ) → f : ( x , x ) → f : ( x , x ) → f : ( x , x ) → f : ( x , x ) → f : ( x , x ) → f : ( x , x ) → f : ( x , x ) → f : ( x , x ) → f : ( x , x ) → f : ( x , x );( x , x ) → f : ( x , x ) . Examples 6 and 7 show how functions f : X k → X that are II- n for n ≥ may be involutory ofsome order, but the involutory order is sensitive to the cardinality of X , unlike II functions. A natural question to consider is what the interpretation is of functions that are n -involutory in termsof the recurrence relations that they represent. We can approach this question more generally in lightof a far less restrictive condition than being n -involutory: Definition 4.
A function f : X k → X is n -involutory at point x ∈ X k when f n ( x ) = x . n -involutory at some point reveal specific cycles of recurrence relations that theyrepresent. For instance, the function in Example 3 is ( k + 1) -involutory, and relatedly, any seed valuein C k generates a ( k + 1) -cycle of the recurrence relation represented by it, a n + k = A − (cid:80) k − i =0 a n + i .This relationship is to be expected, since the iterates of a function f : X k → X encode every k termsof the recurrence relation it represents. The relationship is formalized in the following claim: Claim . If a recurrence relation given by (4) has a j -cycle, then f is jgcd ( j,k ) -involutory at a point. If f is n -involutory at a point, then the recurrence relation has a j -cycle for some j such that j | n .In the special case in which a recurrence relation given by (4) has a k − cycle characterized by some x ∈ X k , then f is -involutory at x ; that is, x is a fixed point of f . By definition, such a fixedpoint x = ( x , x , ..., x k ) ∈ X k satisfies f ( x , x , ..., x k ) = x f ( x , x , ..., x k , x ) = x ... f ( x k , x , x , ..., x k − ) = x k . Conversely, such a fixed point of f corresponds with a j -cycle of the recurrence relation (4), where j | k . In the special case in which the recurrence relation has a 1-cycle, so that the recurrence relationis constant when seeded by the element x ∈ X of the 1-cycle, then the corresponding fixed point of f , x ∈ X k , is symmetric in the sense that x i = x ∀ i ∈ { , , ..., k } . Of course, if f is a symmetricfunction, then all the fixed points of f must be symmetric as such and the recurrence relation itrepresents can only have 1-cycles.While Claim 1 asserts that j -cycles in a recurrence relation correspond with its representativefunction f : X k → X being involutory of some order at a point, in fact, any such j -cycle can beunderstood as corresponding with a point at which a function is -involutory even when j (cid:45) k , by atrivial redefinition of f that augments its domain. By defining the map ˜ f : X k (cid:48) → X for k (cid:48) > k suchthat j | k (cid:48) , given as ˜ f : ( x , x , ..., x k (cid:48) ) → f (˜ x k (cid:48) − k +1 , ˜ x k (cid:48) − k +2 , ..., ˜ x k (cid:48) ) | ˜ x m = f (˜ x m − k , ˜ x m − k +1 ,..., ˜ x m − ) for m ≥ k +1;˜ x m = x m for m ≤ k , then a j -cycle of the recurrence relation represented by f corresponds with a fixed point of ˜ f . Forinstance, the recurrence relation in Example 3 that was characterized by f : C k → C —and for which13very point in C k +1 is a ( k + 1) -cycle— satisfies a n + k +1 = ˜ f ( a n , a n +1 , ..., a n + k ) , where ˜ f : C k +1 → C is defined as ˜ f : ( x , x , ..., x k +1 ) → A − ˜ x k +1 − k (cid:88) j =2 x m | ˜ x k +1 = A − (cid:80) kq =1 x q = x . Thus, ˜ f is simply ˆ id C k +1 (as defined in Lemma 2), which is, of course, -involutory at every point in C k +1 . In general, we can thus understand attractors of a recurrence relation (4) through fixed pointiteration of the self-map f given function f (appropriately augmented) that characterizes behavior ofthe recurrence relation. This paper offers an introduction to a proposed notion of defining function iteration for maps f : X k → X that is based on representing recurrence relations of the kind given in (4). There are severalquestions that can be examined based on the definitions posed in section 3 that are not considered inthis paper but that can likely be addressed within the scope of combinatorics and group theory. Forinstance, for finite sets X , one can consider enumerating the possible functions that are II- n { j } for aspecified number of arguments j (and possibly also involutory of some integral order) and determiningwhen they are symmetric. This problem extends the line of research started in 1800 by HeinrichAugust Rothe, who obtained a recursive formula defining the telephone numbers , which enumerate theinvolutions in the symmetric groups (c.f. Chowla et al. (1951) and Knuth (1973)). Moreover, as perthe fifth property of Lemma 2, one may consider the conjugacy classes of functions that are involutoryof some order, as well as the class number associated with various sets.Likewise, one may also consider how the second claim in Lemma 3 may extend to II- n functions.We can heuristically consider a possible extension by further examining Example 4. Observe that thefunction in Example 4 can afford to be asymmetric while still satisfying the restriction of being II-3because there are fewer than three arguments ( k = 2 ). Suppose more generally that f in this setup isconstrained to be II- n for integer n ≥ and is defined as a linear map in k arguments, i.e. f : C k → C ,where f : ( x , x , ..., x k ) → (cid:80) ki =1 a i x i . Note that all coefficients a i must be n th roots of unity but nonecan be unity itself, allowing the coefficients n − degrees of freedom. However, if k ≥ m ( n −
1) + 1 for an integer m ≥ , one can see that at least m + 1 coefficients must be identical by the pigeonholeprinciple. One can thus consider more generally if a function f : X k → X that is II- n for integer n ≥ k ≥ m ( n −
1) + 1 must be symmetric in at least m + 1 of its arguments.Another question that may be considered is how Proposition 2 might be extended. Examples 6 and7 demonstrate functions defined on finite sets that are II- n (and symmetric) and violate Proposition2 insofar as not having a common involutory order (depending only on k ). Nonetheless, they areinvolutory of some order. One can thus consider for finite sets X under what conditions functionsthat are II- n are involutory of some order, and how this order varies with the cardinality of X . Onemay also consider how many distinct kinds of cycles are associated with such functions (e.g. recall allelements in X in Example 6 were part of one of two 4-cycles or a singleton cycle). Moreover, whileProposition 2 gives sufficient conditions for being n -involutory, one may consider what the necessaryconditions are, including whether there are examples of such functions that are asymmetric outside ofthe kind described by property 4 of Lemma 2.A final point to consider is whether we may likewise suitably define function iteration for functions f : X k → X (cid:96) . The multiplicity of outputs may be reasonably treated as representing distinct recursiveequations, but there is no unique suitable way to treat the inputs in this case. For instance, we maydefine function iteration of a function f : X → X in a way that represents the recursive system a n +2 = f ( a n , b n , a n +1 ) b n +2 = f ( a n , b n , b n +1 ) , or alternatively a n +2 = f ( a n , b n , a n +1 ) b n +2 = f ( a n , b n , a n +1 ) , or in terms of other combinations of three inputs. The state of the first system is characterized by ( a n , a n +1 , b n , b n +1 ) and function iterates may thus be defined as a self-map over X , while the stateof the second system is characterized by ( a n , a n +1 , b n ) and function iterates may thus be defined asa self-map over X . Consequently, it would be more appropriate to define function iteration in suchcases in a way that suits the particular application considered.15 eferences
Aczel, J. (1948). A Remark on Involutory Functions.
The American Mathematical Monthly 55 (10),638–639.Babbage, C. (1815). An Essay towards the Calculus of Functions.
Philosophical Transactions of theRoyal Society of London 105 , 389–423.Babbage, C. (1816). An Essay towards the Calculus of Functions. Part II.
Philosophical Transactionsof the Royal Society of London 106 , 179–354.Babbage, C. (1820).
Examples of the Solutions of Functional Equations . J. Smith, printer to theuniversity; and sold by J. Deighton & sons, Cambridge.Babbage, C. and Gergonne (1821-1822). Analise algébrique. Des équations fonctionnelles.
Annales demathématiques pures et appliquées 12 , 73–103.Baron, K. and W. Jarczyk (2001). Recent results on functional equations in a single variable, perspec-tives and open problems.
Aequ. math 61 , 1–48.Bogdanov, J. S. (1961). The Functional Equation x n = t . Doklady Akademii Nauk BSSR 5 , 235–237.Chowla, S., I. N. Herstein, and W. K. Moore (1951). On Recursions Connected With SymmetricGroups I.
Canadian Journal of Mathematics 3 , 328–334.Ewing, G. M. and W. R. Utz (1953). Continuous Solutions of the Functional Equation f n ( x ) = f ( x ) . Canadian Journal of Mathematics 5 (1), 101–103.Homs-Dones, M. (2020). A generalization of the Babbage functional equation.
Discrete & ContinuousDynamical Systems - A .Isaacs, R. (1950). Iterates of Fractional Order.
Canadian Journal of Mathematics 2 , 409–416.Knuth, D. E. (1973).
The Art of Computer Programming , Volume 3. Addison-Wesley.Kuczma, M., R. Ger, and B. Choczewski (1990).
Iterative Functional Equations . Number v. 32 inEncyclopedia of Mathematics and Its Applications. Cambridge University Press.McShane, N. (1961). On the Periodicity of Homeomorphisms of the Real Line.
The American Mathe-matical Monthly 68 (6), 562–563.Łojasiewicz, S. (1951). Solution Générale De l’Équation Fonctionelle f ( f ( · · · f ( x ) · · · )) = g ( x ) . Ann.Soc.Polon.Math 24 , 88–91. 16itt, J. F. (1916). On Certain Real Solutions of Babbage’s Functional Equation.
Annals of Mathe-matics 17 (3), 113–122.Vincze, E. (1959). Über Die Charakterisierung Der Assoziativen Funktionen Von Mehreren Veränder-lichen.