Reversibility in the diffeomorphism group of the real line

An element of a group is said to be reversible if it is conjugate to its inverse. We characterise the reversible elements in the group of diffeomorphisms of the real line, and in the subgroup of order preserving diffeomorphisms.


Introduction
An element of a group is reversible if it is conjugate to its inverse.We say that such an element is reversed by its conjugator.A diffeomorphism of the real line R is an infinitely differentiable homeomorphism of R whose derivative never vanishes.We consider the group Diffeo(R) of all diffeomorphisms of R, and the subgroup Diffeo + (R) of order preserving diffeomorphisms.The object of this paper is to characterise the reversible elements in each of these two groups.
An involution in a group is an element of order two.One way to obtain a reversible element is to form the product of two involutions.Such an element is reversed by each of the two involutions, and conversely, an element that is reversed by an involution can be expressed as a product of two involutions.Elements reversed by involutions are called strongly reversible.The only involution in Diffeo + (R) is the identity map.There are many non-trivial involutions in Diffeo(R), but all are conjugate to the map x → −x.
Interest in reversibility originates from the theory of time-reversible symmetry in dynamical systems, and background to the subject can be found in [2,6].Finite group theorists use the terms real and strongly real instead of reversible and strongly reversible, because an element g of a finite group G is reversible if and only if each irreducible character of G takes a real value when applied to g. Reversibility in the homeomorphism group of the real line has been considered before by Jarczyk [4] and Young [12].See also [3,8].Reversibility in the group of invertible formal power series was considered by O'Farrell in [9].Previously, in [1], Calica had studied reversibility in groups of germs of homeomorphisms and diffeomorphisms that fix 0. Reversibility in the diffeomorphism group of the real line is of particular interest because, whilst it is difficult to fully classify conjugacy in Diffeo + (R) and Diffeo(R), we are able to give a complete account of reversibility.
The main results of the paper follow.
Theorem 1.1.An element of Diffeo + (R) is reversible if and only if it is conjugate to a map f in Diffeo + (R) that fixes each integer and satisfies Theorem 1.1 gives us an explicit method to generate all order preserving reversible diffeomorphisms.
The alternatives (i) and (ii) are not exclusive.If f is an order reversing reversible diffeomorphism then Theorem 1.2 tells us that it is strongly reversible.Composing two non-trivial involutions in Diffeo(R) gives rise to an order preserving diffeomorphism; hence f must be an involution.
We prove the following result about composites of reversible maps.
Theorem 1.3.Each member of Diffeo + (R) can be expressed as a composite of four reversible diffeomorphisms.
We do not know whether each element of Diffeo + (R) can be expressed as a composite of three, or even two, reversible elements.We also prove the following result about composites of involutions.
Theorem 1.4.Each member of Diffeo(R) can be expressed as a composite of four involutions.
The number four in this theorem is sharp because each order preserving diffeomorphism that is not strongly reversible cannot be expressed as a composite of three involutions.An obvious corollary of Theorem 1.4 is that each element of Diffeo(R) can be expressed as a composite of two (strongly) reversible diffeomorphisms.
The structure of the paper is as follows.Section 2 contains relevant background material.Then in Section 3 we focus on the group Diffeo + (R), and prove all our results related to that group, including Theorems 1.1 and 1.3.Section 4 is about the group Diffeo(R), and in that section we prove all our results about Diffeo(R), including Theorems 1.2 and 1.4.Finally, in Section 5 we list some open problems.

Background results
An element f of a group G is reversible if there is another element h in G such that hf h −1 = f −1 .We say that h reverses f , or that f is reversed by h.If g is an element of G that is conjugate to f then g is also reversible.We denote the fixed point set of a homeomorphism g by fix(g).Listed below are several results about diffeomorphisms of the real line.
Lemma 2.1 (Sternberg,[11]).Each fixed point free member of Diffeo + (R) is conjugate either to the map x → x + 1 or the map x → x − 1.
We remark that x → x + 1 and x → x − 1 are not conjugate in Diffeo + (R), but they are conjugate by the map x → −x in Diffeo(R).
Lemma 2.2 (Kopell, Lemma 1(a), [5]).Suppose that f and g are C 2 order preserving homeomorphisms of an interval [a, b) such that f g = gf .If g has no fixed points in (a, b), but f has a fixed point in (a, b), then f is the identity map.
For f in Diffeo(R), we use the notation T a f to denote the truncated Taylor series regarded as a formal power series in the indeterminate X.
Let P denote the group of formally invertible formal power series having real coefficients, under the operation of formal composition.The identity of P is the series The formal inverse of a power series P is denoted P −1 .
Lemma 2.3 (Kopell, Lemma 1(b), [5]).Let f and g be two elements of Diffeo + (R) that both fix 0 and commute.If T 0 f = X and 0 is not an interior point of fix(f ), then T 0 g = X also.
Our final lemma is about reversibility of formal power series.We are unable to find a precise reference for this lemma, so we provide a brief proof that relies on results from [7,9].
, where a p = 0.By [9, Theorem 4 and Corollary 6], p is even.Since T 2 commutes with S we deduce from [7, Proposition 1.5] that the X coefficient of T 2 , namely λ 2 , equals 1. Hence λ = −1.Next, we can apply [9, Lemma 3 (ii)] to deduce that either T is an involution (in which case the lemma is proved), or else there is an odd integer q and non-zero real number b q such that T We reach a contradiction because p is even and q is odd.Therefore T is an involution.

Reversible maps.
Elementary dynamical considerations tell us that a reversible element in the group of order preserving homeomorphisms of R must have infinitely many fixed points.In fact, a reversing conjugation can only achieve its purpose by shunting the components of the complement of the fixed point set.This was first pointed out by Calica [1].
We use the following lemma about homeomorphisms.Lemma 3.1.Suppose that f and h are order preserving homeomorphisms of R such that hf h −1 = f −1 .Then each fixed point of h is also a fixed point of f .Proof.Suppose that h fixes the point p.We have two equivalent equations hf h −1 = f −1 and hf −1 h −1 = f .From these equations we obtain hf (p) = f −1 (p) and hf −1 (p) = f (p).Order preserving homeomorphisms such as h have no periodic points other than fixed points; thus The next lemma works for diffeomorphisms, but not for homeomorphisms.Lemma 3.2.Suppose that f and h are order preserving diffeomorphisms of R such that hf h −1 = f −1 .If h has a fixed point then f is the identity map.
Proof.If h is the identity map then f is an order preserving involution, and therefore f is also the identity map.Suppose then that h is not the identity map, but that it nevertheless has a fixed point.Choose a component (a, b) in the complement of fix(h).One of a or b is a real number (that is, we cannot have both a = −∞ and b = +∞).Let us assume that a is a real number; the other case can be dealt with similarly.By Lemma 3.1, f fixes (a, b) as a set.The map f cannot be free of fixed points on (a, b).To see this, suppose, by switching f and f −1 if necessary, that f (x) > x for each real number x in (a, b).Then which is a contradiction.Since f has a fixed point in (a, b), Lemma 2.2 applied to the maps f and h 2 shows that f coincides with the identity map on (a, b).We already know that f coincides with the identity map on fix(h); thus f is the identity map.
We are now in a position to prove the first main result.
Proof of Theorem 1.1.Let t be the map given by t(x) = x + 1 for each x.Then (1.1) states that tf t −1 = f −1 .Thus a diffeomorphism f that satisfies (1.1) is reversible, and likewise all conjugates of f are reversible.
Conversely, suppose that g and h are elements of Diffeo + (R) such that hgh −1 = g −1 .The theorem holds when g is the identity, so let us suppose that g is not the identity.This means that we can assume, by Lemma 3.2, that h is free of fixed points.Observe that h −1 gh = g −1 ; hence by replacing h with h −1 if necessary we can assume that h(x) > x for each x.
By Lemma 2.1 we see that there is an element k in Diffeo + (R) such that khk −1 = t.Define f = kgk −1 .Then tf t −1 = f −1 ; that is, (1.1) holds.Now f , like g, must have a fixed point, and by conjugating f by a translation we may assume that this fixed point is 0. Since translations commute, this conjugation does not affect (1.1).Finally, from the equation tf n t −1 = f −n we deduce that f fixes each integer.
We can construct a diffeomorphism f that satisfies (1.1) explicitly by defining f on [0, 1] to be an arbitrary order preserving diffeomorphism of [0, 1] such that T 0 f = (T 1 f ) −1 , and then extending the domain of f to R using (1.1).More precisely, we have the following corollary of Theorem 1.1.
Remark 3.4.Each map f of part (ii) commutes with x → x + 2. Hence f is the lift under the covering map x → exp(πix) of the order preserving diffeomorphism of the unit circle f given by f (e iπθ ) = e iπf (θ) .Moreover f is reversed by x → x + 1; thus f is reversed by rotation by π.

Composites of reversible maps.
Lemma 3.5.Each fixed point free element of Diffeo + (R) can be expressed as a composite of two reversible elements of Diffeo + (R).
Proof.By Lemma 2.1 it suffices to find a single fixed point free map that can be expressed as a composite of two reversible diffeomorphisms.Let f be a reversible order preserving diffeomorphism such that f (x + 1) = f −1 (x) + 1 for each real number x, and f (y) > y for each element y of (0, 1).The graph of such a map f is shown in Figure 1.
Let a be an element from the interval 1 2 , f 1

2
. Notice that every order preserving diffeomorphism h of 1  2 , a satisfies h(x) < f (x) for x ∈ 1 2 , a .Choose an order preserving diffeomorphism g of a, 5  2 such that T a g = T5 2 g = X, and such that g(x) < f (x) for each x ∈ a, 5  2 .(This construction is possible by a classic result of Borel, which says that to each formal power series P there corresponds a smooth function f defined in a neighbourhood of 0 such that T 0 f = P .)Next, choose an order preserving diffeomorphism k from 1  2 , a to a, 5 2 such that T1 We extend the definition of g to R by defining g(x) = k −1 g −1 k(x) for x ∈ 1 2 , a , and g(x + 2) = g(x) + 2 for all x ∈ R. We extend the definition of k by defining k(x) = k −1 (x) + 2 for x ∈ a, 5  2 and k(x + 2) = k(x) + 2 for all x ∈ R. The resulting maps g and k are both order preserving diffeomorphisms.Moreover, one can check that the equation g(x) = k −1 g −1 k(x) is satisfied for points x in 1  2 , 5 2 .Since both maps commute with x → x + 2, this equation is satisfied throughout R. Finally, we have defined g such that f (x) > g(x) for elements x of 1 2 , 5 2 , and in fact f (x) > g(x) everywhere, again, because both maps commute with x → x + 2. Therefore g −1 f is a fixed point free diffeomorphism expressed as a composite of two reversible maps.
Proof of Theorem 1.3.Choose f in Diffeo + (R).Choose a fixed point free diffeomorphism g such that g(x) < f (x) for each x ∈ R. Then g −1 f (x) > x for each x in R, so the map h = g −1 f is also free of fixed points.Since f = gh, the result follows from Lemma 3.5.
We do not know whether each element of Diffeo + (R) is the composite of three reversible elements.

Order reversing reversible maps.
We denote the set of order reversing diffeomorphisms of R by Diffeo − (R).The next proposition fails for homeomorphisms.

Proposition 4.1. An order reversing member of Diffeo(R) is reversible in Diffeo(R) if and only if it is an involution.
Proof.Involutions are all reversible by the identity map.Conversely, suppose that f ∈ Diffeo − (R), h ∈ Diffeo(R), and hf h −1 = f −1 .By replacing h with hf if necessary, we may assume that h preserves order.From the equation hf = f −1 h we deduce that h fixes the unique fixed point of f .Now, hf 2 h −1 = f −2 , and f 2 preserves order; therefore Lemma 3.2 applies to show that f 2 is the identity map, as required.Proposition 4.1 accounts for all order reversing reversible diffeomorphisms.In Theorem 1.1 we described all order preserving diffeomorphisms that are reversed by order preserving maps.That leaves only order preserving diffeomorphisms that are reversed by order reversing maps.These are examined next.Proof.Let f ∈ Diffeo + (R), h ∈ Diffeo − (R), and hf h −1 = f −1 .We wish to show that f is strongly reversible.Given Lemma 4.2, we may assume that f has a fixed point.By conjugation, we can assume that the fixed point of h is 0. Notice that h permutes the fixed points of f .We define an involutive homeomorphism k by First, suppose that f coincides with the identity on a neighbourhood of 0. In this case we have freedom to adjust the definition of k near 0 so that it is an involutive diffeomorphism, without disturbing the validity of the equation kf k = f −1 .
Second, suppose that 0 is not an interior fixed point of f , but that T 0 f = X.Since h 2 commutes with f , it follows from Lemma 2.3 that T 0 h is an involution, so that k is already a diffeomorphism.Third, suppose that 0 is a fixed point of f and T 0 f = X.By Lemma 2.4, T 0 h is an involution, and again, k is a diffeomorphism.Now suppose that 0 lies inside a component (a, b) of R \ fix(f ).Since f has a fixed point, we know that (a, b) = R.Moreover, because the order reversing map h fixes (a, b), both end points a and b are finite, and h(a) = b and h(b) = a.Therefore h 2 fixes a, b, and 0, and commutes with f .By Lemma 2.2, This means that h and h −1 coincide inside (a, b), so that k is a diffeomorphism, and Proof of Theorem 1.2.Combine Propositions 4.1 and 4.3.
Since all non-trivial involutions in Diffeo(R) are conjugate to x → −x we have the following explicit method to construct all strongly reversible elements of Diffeo(R): Note that the graph of a map reversed by x → −x is symmetric in the line y = −x.
Refer to [4,8,12] for more information on strong reversibility of homeomorphisms.Proposition 4.3 shows that elements of Diffeo + (R) that are reversed by order reversing elements of Diffeo(R) are strongly reversible in Diffeo(R).There are, however, elements of Diffeo + (R) that are reversed by order preserving elements of Diffeo(R) that are not strongly reversible in Diffeo(R).In fact, for order preserving diffeomorphisms, the properties of being reversible in Diffeo + (R) and strongly reversible in Diffeo(R) are logically independent.To demonstrate this, we must, in turn, find an example of an order preserving diffeomorphism that is (i) neither reversible in Diffeo + (R) nor strongly reversible in Diffeo(R); (ii) not reversible in Diffeo + (R), but strongly reversible in Diffeo(R); (iii) reversible in Diffeo + (R), but not strongly reversible in Diffeo(R); (iv) reversible in Diffeo + (R) and strongly reversible in Diffeo(R).
Examples of (i) and (ii) are readily constructed.For (ii), any non-trivial strongly reversible diffeomorphism which coincides with the identity map outside a compact set will suffice, because Theorem 1.1 tells us that such a map cannot be reversible in Diffeo + (R).We now give an example of (iii), and then a non-identity example of (iv).

Example (iii).
We shall describe an order preserving diffeomorphism f that is reversible by order preserving diffeomorphisms, but not by order reversing involutions.The map described is not even strongly reversible as a homeomorphism.We assume some common knowledge of conjugacy in the homeomorphism group of the real line, which can be found, for example, in [3].
We shall define f to be an element of Diffeo + (R) such that fix(f ) = Z.To specify f up to topological conjugacy, it remains only to describe the signature on R \ Z, which we represent by an infinite sequence of + and − symbols.A + symbol corresponds to an interval (n, n + 1) for which f (x) > x for each x ∈ (n, n + 1), and a − symbol corresponds to an interval (n, n + 1) for which f (x) < x for each x ∈ (n, n + 1).The signature of a homeomorphism of R is discussed in more detail in [3].Suppose the signature of f consists of the 12 symbol sequence +, +, +, −, −, +, −, −, −, +, +, −, repeated indefinitely, in both directions.The map f can be chosen to be a diffeomorphism.A portion of a graph of such a function is shown in Figure 2. It satisfies hf h −1 = f −1 , where h is given by the equation h(x) = x + 6.On the other hand, it is straightforward to see (or refer to [3]) that f is not reversible by a non-trivial involution, as the doubly infinite sequence generated by (4.1) read forwards is different from the same sequence read backwards.Then τ is an involutive element of Diffeo − (R), so that f = −τ is an element of Diffeo + (R) that is strongly reversible in Diffeo(R).
On the other hand, τ (x + 2) = −τ (−x) − 2 for all x ∈ R, so Hence f is also reversible in Diffeo + (R).The corresponding result for homeomorphisms is due to Fine and Schweigert [3].

Open questions
We list two open problems which have emerged from our study.
Question 5.1.What is the smallest positive integer m such that each member of Diffeo + (R) can be expressed as a composite of m reversible maps?

4. 2 .Proposition 4 . 3 .
Strongly reversible maps.Lemma 4.2.Fixed point free diffeomorphisms are strongly reversible.Proof.A fixed point free diffeomorphism is, by Lemma 2.1, conjugate in the group Diffeo(R) to x → x + 1, and this map is reversed by the involution x → −x.If a member of Diffeo + (R) is reversed by a member of Diffeo − (R) then it is strongly reversible.

Corollary 4 . 4 .
Let g ∈ Diffeo + (R).Then the following two conditions are equivalent:(i) The map g is reversible in Diffeo(R) by an order reversing diffeomorphism.(ii) There exist (a) a formally invertible power series P that is strongly reversed by the power series −X; (b) a point p and an order preserving diffeomorphism φ : [p, ∞) → [−p, ∞) such that T p φ = P ; (c) h ∈ Diffeo(R), such that g = hf h −1 , where