1,721,108 research outputs found
Critical functions for complex analytic maps
No full textCritical functions measure the width of the domain of stability around a given fixed point or an invariant circle for complex analytic and area-preserving maps. The author discusses their dependence on the rotation number of the invariant curves and proposes some new methods to determine them based on the existence of critical points and on some properties of quasiconformal maps. By means of the majorant series method some rigorous estimates are given for complex area-preserving maps like the semistandard map and the modulated singular map. In particular, the author makes use of the Brjuno function to interpolate critical maps and proves that the convergence of the Brjuno function is a necessary and sufficient condition for the existence of an analytic invariant curve of a given rotation number. The author also discusses the optimality of the rigorous bounds obtained
A Method for Accurate Stability Bounds in a Small Denominator Problem
No full textThe author considers the problem of obtaining realistic lower bounds for the Siegel radius. Recent advances of the analysis of Siegel disks allows one to give a very accurate numerical algorithm based on rigorous results. He finds that for non-quadratic polynomial maps the maximal Siegel radius might correspond to rotation numbers different from the golden mean
Some arithmetical aspects of renormalization in Teichmüller dynamics : on the occasion of Corinna Ulcigrai winning the Brin Prize
No full textWe present some works of Corinna Ulcigrai closely related to Diophantine approximations and generalizing classical notions to the context of interval exchange maps, translation surfaces and Teichmuller dynamics.We present some works of Corinna Ulcigrai closely related to Diophantine approximations and generalizing classical notions to the context of interval exchange maps, translation surfaces and Teichmuller dynamics
The solar system between order and chaos
No full textPoincaré’s fertile error reintroduced the oldest open question in dynamical systems: the problem of stability of orbits in the n-body problem. After Poincaré the problem was studied by Birkhoff, Kolmogorov, Arnold and Moser. Laskar, using the secular system introduced by Lagrange, showed that the orbits of the planets in a realistic model of the solar system are chaotic with a Lyapounov exponent of 1/(5 Myrs)
Diophantine Conditions and Estimates for the Siegel Radius: Analytical and Numerical results
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