On hyperbolicity in the renormalization of near-critical area-preserving maps (Q727423)

This is a rigorous numerical investigation of some universal quantities associated to MacKay's renormalization operator for pairs of area-preserving maps near the fixed point found in work of \textit{G. Arioli} and the author [Commun. Math. Phys. 295, No. 2, 415--429 (2010; Zbl 1196.37074)]. It is shown that a suitable extension of an iterate of the renormalization operator restricted to commuting pairs with zero Calabi invariant is hyperbolic with a fixed point having a single expanding direction. Among other quantities, a numerical approximation for the expanding eigenvalue is found with precise bounds. The methods make essential use of estimates carried out by computer.