On Fixed Points of Commuting Functions

On Fixed Points of Commuting FunctionsAuthor(s): Haskell CohenSource: Proceedings of the American Mathematical Society, Vol. 15, No. 2 (Apr., 1964), pp.293-296Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/2034056 .

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ON FIXED POINTS OF COMMUTING FUNCTIONS'

HASKELL COHEN

There is a rather well-known conjecture that if f and g are continuous functions on [0, 1] to itself which commute (i.e., f(g(x)) =g(f(x))), then they have a common fixed point. The conjecture is apparently due independently to Eldon Dyer and Allen Shields, and has been generalized by J. R. Isbell [2]. The conjecture is easily verified for polynomials f and g by referring to some work of J. F. Ritt [3] who showed that the only commuting polynomials, aside from some trivial cases are the Tchebycheff polynomials all of which have a common fixed point. This result is stated more explicitly by Block and Thielman [I].

The author has noted that certain functions with broken line graphs, e.g.,

{2x if 0 < x < f(x) 1

i2- 2x if I <x_ 1

and

'3x if 0 < x <

g(x) = .2 -3x if < x < 23

3x-2 if 23 < x_ 1

also commute, and that these share with Tchebycheff polynomials (suitably modified by Lemma 1 so that they take [0, 1] into [0, 1]) the property he calls fullness. (A function on [0, 1 ] to itself is full if the interval may be subdivided into a finite number of subintervals on each of which the function is a homeomorphism onto [0, 1].) In fact every pair of nontrivially commuting continuous functions known to the author are either full, or it is possible to find a subinterval which the restrictions of the functions take onto itself and on which (with the scale properly changed) they are full. The result shown in this note is that if two full functions commute, they have a common fixed point. It is hoped that the result and/or some of the lemmas will be useful in studying the general problem.

DEFINITION. Two functions f and g defined on a set X to itself are said to commute if for each xCX we have f(g(x)) = g(f(x)); f and

Received by the editors July 6, 1962 and, in revised form, December 15, 1962. 1 This work was partially supported by the National Science Foundation through

grant NSF-G-14085.

293



294 HASKELL COHEN [April

g have a common fixed point if there is an x EX such that f(x) = g(x) = x. We use juxtaposition of functions to indicate composition and may write the commuting property as fg = gf.

LEMMA 1. If f and g are functions on the interval [a, b] to itself and h is a homeomorphism of [a, b] onto [c, d], then hfh-' and hgh-lare functions on [c, d] to itself which commute and have a commonfixed point if and only if f and g commute and have a common fixed point.

The proof of this lemma is straightforward and is omitted.

LEMMA 2. If there are continuous commuting functions on [0, 1 ] to itself without a common fixed point, then there are also onto functions with these properties.

PROOF. Suppose f and g satisfy the hypotheses of the lemma. Let a,=max(inff, inf g) and bi=min(supf, sup g). Since f and g commute, their ranges intersect, and, a, < bi. Let fi and g1 be f and g restricted to [a,, bi], respectively; fi and gi take [a,, bi] into [a,, bi ] for if, for example, f1(x) > bi, there is yE [0, 1 ] such that g(y) = x, and gf(y) =fg(y) =f(x) > bi implies bi <min(supf, sup g). Inductively we let ai = max(inf fi-i inf g4-1), bi = min(supfi-i, sup gi-1), fi=fi-11 [ai, bi] and gi=gi-1 [as, bj]. The set { [as, bi] } forms a nested sequence of closed intervals and has a nonnull intersection. If the intersection were degenerate, f and g would have a common fixed point; hence, the intersection is an interval [a, b ], and f =f I [a, b ] and g=gg [a, b] are onto [a, b]. Now letting h be a homeomorphism of [a, b] onto [0, 1] and using Lemma 1, we get hfh-1 and hgh-1 as the required functions.

LEMMA 3. If f and g are commuting functions, then f and gf are commuting functions which have a common fixed point if and only if f and g have.

The proof of this lemma is also quite trivial and hence omitted. DEFINITION. A continuous function f: [0, 1 ] [0, 1] will be called

full if there exists a partition Pf = {Xoi Xo, .. * Xn } of [0, 1] with xo = 0, xi <xi+, and xn =1 such that for each i we have fI [xi, xi+,] is a homeomorphism onto [0, 1].

It is immediate that the composition of two full functions is full and that Pf is unique.

DEFINITION. A partition Pf is regular if its subintervals are all the same length. A partition Pv refines Pf uniformly if each Pf interval is the union of the same number of Pv intervals.



I964] ON FIXED POINTS OF COMMUTING FUNCTIONS 295

LEMMA 4.2 If fi and gi are commuting full functions without a common fixed point, there are functions f and g having the same properties and in addition are such that f(O) =g(1) =0, f(l) =g(0) = 1, Pf, PF, and Pf0 are regular, and P, refines Pf uniformly.

PROOF. Since fi(0) = gi(0) = 0 guarantees a common fixed point, we need only (after possibly renaming the functions) consider the cases (1)fi(O)=0;g1(0)=1 and (2) f (0) = 1=gi(O). In case (1) fi(1) =fig1(0) =gif1(0) =gi(O) = 1; hence gi(l) must be 0 else 1 is a common fixed point. In this case let f2=fl and g2=gl. In case (2) fi(1)=figi(0) =gif1(O)=g,(1); therefore to avoid a common fixed point we must have fl(1) = gi(l) = 0. In this case we let f2=f1gl and g2= gl. Now f2(0) =figl(0) =fl(l) = 0, g2(0) = g1(0) = 1, f2(1) =f1gj(1) =f,(0) = 1 and g2(1) = g1(l) =0. In either case let f3 =f2 and g3 = g2f2. Clearly Pg3 refines Pf3 uniformly, and similarly Pf3g3 refines Pg3 uniformly. Now let 4 be any order preserving homeomorphism on [0, 1] taking Pf 2,3 into the corresponding regular partition of [O, 1]. Let f=4f/30-' and g = 0g3f-'. It is easy to verify that these functions have the required properties.

THEOREM. Commuting fullfunctions must have a common fixed point.

PROOF. If not, there exist f and g satisfying Lemma 4. Suppose Pf= {0, 1/n, 2/n, * * *, 1} and P= {0, l/m, 2/m, - * *, 1}; then Pfg {0, l/mn, 2/mn, * , 1 } and m and n are odd. We adopt the notation that fi=fI [(i-1)/n, i/n] and gi=gI [(i-1)/m, i/im], let r= (n+1)/2, s= (m+ 1)/2, and consider the case when r is odd and s is even (similar arguments can be made for the other cases). Note that D(figj) (the domnain of figj) for each i and j is some subinterval of Pf1 and, in particular,

-r- r -1 1 - D(gifr) = [ - 1 1

_ n n mn_

[r-1 1 r-1 21 D(g2fr)= + +_ * *+ I

n mn n mnj

Fr-1 s-1 r-1 sl -mn-i mn +1 D(gjfr)= + ' +_ =

L n m n m _ 2mn 2mn

X The author is indebted to the referee for the statement and the shortened proof of this lemma.



296 HASKELL COHEN

Similarly

Fs-I s -i 2 D(figL) = +, + m,

_m tm mn

-S-1 1 s-~1 2 D(f2gs) = + _ +. ,**

_ mn m rnn

r-i r-l s-i rl Fmn-i mn +11 D(frgs) = +

r- r + n _ n J

_ m n m nJ 2mn 2mn

and we have shown that D(g,fr) D(frg.). Now g. is continuous and onto [0, 1 ]; so its graph must intersect the diagonal of [0, I ] X [0, 1 ] and g. has a fixed point zi. Since D(g.) CD(fr), z,ED(f,) and thus ziED(f7g.) = D(gjf). Therefore g.fj(z1) =frg.(z) =fr(zi) and Z2 =fr(Zl)

is a fixed point of g.. Continuing we get a sequence { zV, } of fixed points of g. where z,+l =fr(z,). Since fr is monotone, the sequence { zp } con- verges to, say, z which clearly is both a fixed point of f and g. This contradiction completes the proof.

REFERENCES

1. H. D. Block and H. P. Thielman, Commutative polynomials, Quart. J. Math. Oxford Ser. (2) 2 (1951), 241-243.

2. J. R. Isbell, Commuting mappings of trees, Bull. Amer. Math. Soc. 63 (1957), 419.

3. J. F. Ritt, Permutable rationalfunctions, Trans. Amer. Math. Soc. 25 (1923), 399-448.

LOUISIANA STATE UNIVERSITY



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On Fixed Points of Commuting Functions