A very brief history of universality in period doubling

In 1976, M.J. Feigenbaum got me interested in his Aug 1975 discovery of universality in one-dimensional iterative maps. The first published report on this work is dated Aug 1976 (Los Alamos Theoretical Division Annual Report 1975-1976, pp. 98-102, read it here), and it became rapidly widely known through many seminars given by Feigenbaum, both in US and Europe. His first paper, submitted to Advances in Mathematics in Nov 1976 was rejected. The second paper was submitted to SIAM Journal of Applied Mathematics in April 1977 and rejected in October 1977. Finally, J. Lebowitz published both papers without further referee pain (M. J. Feigenbaum, J. Stat. Phys. 19, 25 (1978) and 21, 6 (1979)). By 1978 others have published similar results, and by 1979 mathematicians also understood that the numerical methods we used to solve the universal equations were in fact convergent. They did the usual; they attached various names to the equations, and they changed letters around to make the equations unintelligible to physicists. Fortunately the re-lettering did not stick.

Following Feigenbaum's functional formulation of the problem, in spring 1976 I derived the equation for the period doubling fixed point function (not a big step - it is the limit of his functional recursion sequence), which has since played a key role in the theory of transitions to turbulence. Since then we have generalized the universal equations to period n-tuplings; constructed universal scaling functions for all winding numbers in circle maps, and established universality of the Hausdorff dimension of the critical staircase.

Thanks to my diaries we can be ridiculously precise about the dates. Mitchell told me about existence of period doubling universality at 10pm on December 19 1975, in a bar in New York City, over a glass of good red wine and in presence of my Black Irish Poetess. I started thinking about functional equations on May 1, 1976, and wrote down the period doubling fixed point function equation on May 3, 1976.

What came next? We can quote Freeman Dyson: As usual when you discover something new, the response comes in three waves. First, this is nonsense. Second, this is trivial. Third, this is important, and we did it before you did.


Predrag Cvitanović, Atlanta, January 23, 2013