Statistical properties of Poincaré recurrences and Afraimovich-Pesin dimension for the circle map

Statistical characteristics of Poincaré recurrences are studied numerically for the linear and nonlinear circle map with different irrational values of the rotation number. It is first established that the dependence of the minimal Poincaré return time on the size of a return region has several universal properties. The theoretical result for the Afraimovich-Pesin dimension equality αc=1 is confirmed for Diophantine irrational numbers in both the linear and nonlinear circle map. It is shown that the gauge function 1/t cannot be used for Liouvillian numbers. We also show that the set generated in a stroboscopic section of the dynamics of a nonautonomous oscillator possesses the same basic features that are obtained for the circle map.