Regularity Properties of Critical Invariant Circles of Twist Maps, and Their Universality

Abstract

We accurately compute the golden and silver critical invariant circles of several area-preserving twist maps of the cylinder. We define some functions related to the invariant circle and to the dynamics of the map restricted to the circle (for example, the conjugacy between the circle map giving the dynamics on the invariant circle and a rigid rotation on the circle). The global Hölder regularities of these functions are low (some of them are not even once differentiable). We present several conjectures about the universality of the regularity properties of the critical circles and the related functions. Using a Fourier analysis method developed by de la Llave and one of the authors, we compute numerically the Hölder regularities of these functions. Our computations show that—within their numerical accuracy—these regularities are the same for the different maps studied. We discuss how our findings are related to some previous results: (a) to the constants giving the scaling behavior of the iterates on the critical invariant circle (discovered by Kadanoff and Shenker) and (b) to some characteristics of the singular invariant measures connected with the distribution of iterates. Some of the functions studied have pointwise Hölder regularity that has different values at different points. Our results give convincing numerical support to the fact that the points with different Hölder exponents of these functions are interspersed in the same way for different maps, which is a strong indication that the underlying twist maps belong to the same universality class. In particular, the numerical results on the regularity of the so-called big conjugacies imply that the Hölder spectra of the functions conjugating the dynamics on the critical invariant circle to a rigid rotation are the same. This, in turn, shows that the invariant measures on the critical circles have the same singularity spectra.