First posed in 1954, the problem remained unsolved for more than a decade, during which several mathematicians made incremental progress toward an affirmative answer.
In 1967, William M. Boyce and John P. Huneke independently[1]: 3 proved the conjecture to be false by providing examples of commuting functions on a closed interval that do not have a common fixed point.
A 1951 paper by H. D. Block and H. P. Thielman sparked interest in the subject of fixed points of commuting functions.
[2] Building on earlier work by J. F. Ritt and A. G. Walker, Block and Thielman identified sets of pairwise commuting polynomials and studied their properties.
They proved, for each of these sets, that any two polynomials would share a common fixed point.
[3] Block and Thielman's paper led other mathematicians to wonder if having a common fixed point was a universal property of commuting functions.
are two continuous functions that map a closed interval on the real line into itself and commute, they must have a common fixed point.
The same question was raised independently by Allen Shields in 1955 and again by Lester Dubins in 1956.
[4] John R. Isbell also raised the question in a more general form in 1957.
[5] During the 1960s, mathematicians were able to prove that the commuting function conjecture held when certain assumptions were made about
[6] Gerald Jungck refined DeMarr's conditions, showing that they need not be Lipschitz continuous, but instead satisfy similar but less restrictive criteria.
[13] In his thesis, Boyce identified a pair of functions that commute under composition, but do not have a common fixed point, proving the fixed point conjecture to be false.
[14] In 1963, Glenn Baxter and Joichi published a paper about the fixed points of the composite function
[15] In an independent paper, Baxter proved that the permutations must preserve the type of each fixed point (up-crossing, down-crossing, touching) and that only certain orderings are allowed.
"[2][16][17] His program carefully screened out those that could be trivially shown to have fixed points or were analytically equivalent to other cases.
After eliminating more than 97% of the possible permutations through this process, Boyce constructed pairs of commuting functions from the remaining candidates and was able to prove that one such pair, based on a Baxter permutation with 13 points of crossing on the diagonal, had no common fixed point.
Boyce published a separate paper describing his process for generating Baxter permutations, including the FORTRAN source code of his program.
[18] John P. Huneke also investigated the common fixed point problem for his Ph.D. at Wesleyan University, which he also received in 1967.
In his thesis, Huneke provides two examples of function pairs that commute but have no common fixed points, using two different strategies.
Huneke used this principle to construct sequences of functions that will converge to the counterexample to the common fixed point problem.
[20] Although the discovery of counterexamples by Boyce and Huneke meant that the decade-long pursuit of a proof of the commuting function conjecture was lost, it did enable researchers to focus their efforts on investigating under what conditions, in addition to the ones already discovered, the conjecture still might hold true.
[2] Boyce extended the work of Maxfield/Mourant and Chu/Moyer in 1971, showing weaker conditions that allow both of the commuting functions to have period 2 points but still imply that they must have a common fixed point.
[23] His work was later extended by Theodore Mitchell, Julio Cano, and Jacek R.
[24][25][26] Over 25 years after the publication of his first paper, Jungck defined additional conditions under which