Combination laws for scaling exponents and relation to the geometry of renormalization operators

Renormalization group has become a standard tool for describing universal properties of different routes to chaos – period-doubling in unimodal maps, quasiperiodic transitions in circle maps, dynamics on the boundaries of Siegel disks, destruction of invariant circles of area-preserving twist maps, and others. The universal scaling exponents for each route are related to the properties of the corresponding renormalization operators. We propose a Principle of Approximate Combination of Scaling Exponents (PACSE) that organizes the scaling exponents for different transitions to chaos. Roughly speaking, if the combinatorics of a transition is a composition of two simpler combinatorics, then the scaling exponents of the combined combinatorics is approximately equal to the product of the scaling exponents, both in the parameter space and in the configuration space, corresponding to each of these two combinatorics. We state PACSE quantitatively as precise asymptotics of the scaling exponents for combined combinatorics, and give convincing numerical evidence for it for each of the four dynamical systems mentioned above. We propose an explanation of PACSE in terms of the dynamical properties of the renormalization operators – in particular, as a consequence of certain transversal intersections of the stable and unstable manifolds of the operators corresponding to different transition to chaos.