Renormalization group operators for maps and universal scaling of universal scaling exponents

Renormalization group (RG) methods provide a unifying framework for understanding critical behaviour, such as transition to chaos in area-preserving maps and other dynamical systems, which have associated with them universal scaling exponents. Recently, de la Llave et al. (2007) [10] have formulated the Principle of Approximate Combination of Scaling Exponents (PACSE for short), which relates exponents for different criticalities via their combinatorial properties. The main objective of this paper is to show that certain integrable fixed points of RG operators for area-preserving maps do indeed follow the PACSE. ► Renormalization group operators for invariant circles of area-preserving maps with quadratic irrational rotation number are presented. ► The simple fixed points and two-cycles and their scaling exponents are calculated explicitly. ► It is shown that the simple fixed points follow the Principle of Approximate Combination of Scaling Exponents (PACSE).