A computer assisted proof of universality for cubic critical maps of the circle with Golden Mean rotation number
In order to explain the universal metric properties associated with the breakdown of invariant tori in dissipative dynamical systems, Ostlund, Rand, Sethna and Siggia together with Feigenbaum, Kadanoff and Shenker have developed a renormalisation group analysis for pairs of analytic functions that glue together to make a map of the circle. Using a method of Lanford's, we have obtained a proof of the existence and hyperbolicity of a non-trivial fixed point of the renormalisation transformation for rotation number equal to the golden mean (√5 - 1/2). The proof uses numerical estimates obtained rigorously with the aid of a computer. These computer calculations were based on a method of Eckmann, Koch and Wittwer.