Computational Aspects of Universal Constants for Critical Circle Maps

To explore the universal properties linked to the breakdown of invariant tori in dissipative dynamical systems, Ostlund, Rand, Sethna and Siggia, together with Feigenbaum, Kadanoff and Shenker developed a renormalization group approach for pairs of analytic functions on the unit circle. Based on this, D. Mestel, utilizing a method from Lanford, established the existence of a non-trivial fixed point <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(f_{\star},g_{\star})$</tex> for the renormalization transformation with “golden mean” <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\frac{\sqrt{5}-1}{2}$</tex> rotation number. The numerical results obtained with the help of a computer became the key to this proof. In present paper we find numerically the universal constants of critical circle map associated by pair <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(f_{\star},g_{\star})$</tex>. These constants play an important role in applications of chaotic dynamical systems.