Eric Aurell: Feigenbaum's second theory of the presentation function

Abstract: The understanding of universality in transitions to chaos achieved in the 1970ies in the sense of physics, and later as rigorous
mathematics, is based on renormalization. In the abstract renormalization is a change of the representation of some object from a change of resolution of observation. As applied to dynamics at the transition to chaos the change is from observing the system at every (discrete) time point to more widely spaced time points. The paradigmatic case is period-doubling where the sequence is every time point, every second time point, every fourth time point, and so on.

From the fixed point of renormalization one can compute metric properties, of which one of the most famous is Feigenbaum's quantity \(\alpha\) in period-doubling. Wikipedia says that it is 2.502907875095892822283902873218 (I personally never computed it with so many digits)

This alpha is however just one out of many metric properties which can also be computed from the renormalization fixed point. For instance, it is well known that 2^k almost returns of the critical point (conventionally x=0) to itself at the accumulation point of period doublings scale as (1/-\alpha)^k, the renormalization transformation in fact encodes this regularity. However, it is then obvious that 2^k almost returns of one iteration of the critical point (conventionally f(0)) must scale as (1/\alpha)^(2k). Hence (1/-\alpha) and (1/\alpha^2) both appear as metric regularities, and these numbers can be extended to a real function \sigma(t) on [0,1/2] which Feigenbaum first called 'scaling function' and later 'presentation function'.

A bit less known is the fact that the presentation function can itself be computed as a fixed point, without direct reference to the standard renormalization, or its fixed point.

I will present how this is done, with no pretence of mathematical rigor, and without showing my own numerical data on the problem, which I did at an earlier time in this seminar. I will also not consider the case of golden mean mode locking of circle maps, where a similar analysis has been carried out, but only focus on period-doubling.

The talk is based on M Feigenbaum, J Stat Phys vol 52 Nos 3/4 (1988) pp. 527-569, a paper which has been cited 121 times according to Google
scholar, but where the main idea, to my knowledge, has not been much developed further, nor has it been taken up by others.