Circle maps with symmetry-breaking perturbations
The circle maps under investigation in this paper are of the form \(f_ \Omega(\theta)=\theta+\Omega-(a/2\pi)\sin(2\pi\theta)\) \(- (b/4\pi)\sin(4\pi\theta)\), with \(a,b\geq 0\). For the parameters \(a\) and \(b\), interesting features occur near the critical surface \(a+b=1\), where the map ceases to be a diffeomorphism. The parameter \(\Omega\) is used to control the winding number \(\rho\), which is taken to be either \(\gamma^{-1}\) or \(2\gamma^{-1}-1\), where \(\gamma\) is the golden mean. Universal exponents related to circle maps were first discovered numerically and later explained by renormalization techniques, where they appear as eigenvalues of the linearized renormalization operator around a fixed point. In the present work, four such universal exponents make their appearance. The numerical experiments which produce them rely on symmetry breaking perturbations, whereby it is essential to introduce with \(a=\varepsilon\) and \(b=1-\varepsilon\) a continuous parameter \(\varepsilon\) to control the symmetry violating terms. New crossover properties thus appear, which could not be obtained by discretely changing the universality class. For a rigorous explanation of the four exponents, the authors introduce an extended renormalization transformation, involving a third functional component. The interpretation which emerges is that one of the exponents corresponds to the change in winding number; two others form a degenerate pair corresponding to crossing the critical surface; the fourth and truly new one is somehow associated with antisymmetric perturbations of the circle map.
scientific article; zbMATH DE number 91638
scientific article