Poincaré Recurrences in the Circle Map: Fibonacci Stairs - Anishchenko, V.S.; Semenova, N.I.; Vadivasova, T.E. (Discontinuity, Nonlinearity, and Complexity, 2026)
We show that the dependence of the mimimal Poincaré return time on the vicinity size is universal for the golden and silver ratios in the circle map and can be referred to as the “Fibonacci stairs”. The rigorous result for the Afraimovich-Pesin dimension equality αc = 1 is confirmed for irrational rotation numbers with the measure of irrationality μ = 2. It is shown that some transcendental number are Diophantine and have the measure μ = 2. It is also confirmed that the gauge function 1/t cannot be applied for Liouvillian numbers. All the obtained features hold for both the linear and the nonlinear circle map.