Is there any expression for the Feigenbaum constants ?

Question

It has puzzled me for a long time that the Feigenbaum constant $\delta$ and reduction parameter $\alpha$ do not seem to be related to other constants (that is, numerically), even not to each other. In fact I have never seen them expressed as an integral, any kind of series or product, a nested expression, ... The only thing I have found on the internet is this algorithmic approach which further links to this, but it seems rather like an "a posteriori" method, being interested more in the algorithm than in the nature of $\delta$. Anyway, I would not expect the prominent occurrence of the number $163$ in the algorithm to have a deeper meaning.

On the other hand, I wouldn't be too surprised to see $\delta$ written as a continued fraction with "defineable" terms, given the fact that it can be defined by $\delta=\lim\limits_{n\to\infty}\dfrac{\mu_n-\mu_{n-1}}{\mu_{n+1}-\mu_n}$, where the $\mu_n$ are the bifurcation points of an iterated map. But I have never seen one either. Any leads?

Answer 1

The Feigenbaum constant is the largest eigenvalue of the derivative of the renormalisation operator at its unique fixed point. There is a beautiful article of Lyubich in the October 2000 Notices of the AMS, entitled "The Quadratic Family as a Qualitatively Solvable Model of Chaos", in which he summarises the connection between universality and renormalisation (pages 1046, 1049-1051).

That doesn't directly answer your question, but hopefully that interpretation of the Feigenbaum constant, together with Lyubich's article and/or the references therein, may be of interest.

Answer 2

Did you look at the papers by Keith Briggs referenced in the Wikipedia article about the Feigenbaum constants? They were computational too, but there was math in them.