The Critical Renormalization Fixed Point for Commuting Pairs of Area-Preserving Maps

Renormalization conjecture and rigidity theory for circle diffeomorphisms with breaks

“…This condition is, however, rather artificial in the setting of nonlinear IETs. Without the derivative-matching condition one obtains a circle diffeomorphism T with two break points at x (1) br = 0 and x (2) br = ξ. Since the two break points belong to the same orbit of T , i.e.…”

“…Since the two break points belong to the same orbit of T , i.e. T x (2) br = x (1) br , one can piecewise-smoothly conjugate T to a circle map with a single break point. A natural question to ask is when are two maps of this type smoothly, or piecewise-smoothly, conjugate to each other.…”

“…The idea of renormalization originally came from statistical mechanics, where it provided an explanation for critical phenomena, by classifying systems into different universality classes, according to their scaling limits and corresponding critical exponents. In dynamics, apart from providing an explanation for the universality of infinitely renormalizable unimodal maps, renormalization methods have also led to advances in rigidity theory [4,5,[13][14][15][16][17][18][19]29], complex dynamics [27,30], KAM (Kolmogorov-Arnol'd-Moser) theory [20][21][22], break-up of invariant tori [1,25], and the reducibility of cocycles and skew-product flows [3,23].…”

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

“…Of course, the eventual aim is to find the critical fixed points and the critical scaling exponents. There is already a vast literature about properties of similar RGO, including their critical fixed points, periodic orbits, or the strange attractor set, and the connections with critical phenomena, [1,7,5] but these are not reviewed extensively here since this paper only discusses the simple fixed points.…”

“…[1,2] for a review and further references. Some of the specific examples of the use of RG are in the studies of the breakup of invariant tori of area-preserving maps [3][4][5] and Hamiltonian systems [6,7], transition to chaos through period doubling in maps of the interval [8], and many others.…”