Intrinsic stickiness and chaos in open integrable billiards: tiny border effects.

Rounding border effects at the escape point of open integrable billiards are analyzed via the escape-time statistics and emission angles. The model is the rectangular billiard and the shape of the escape point is assumed to have a semicircular form. Stickiness, chaos, and self-similar structures for the escape times and emission angles are generated inside "backgammon" like stripes of initial conditions. These stripes are born at the boundary between two different emission angles but with the same escape times and when rounding effects increase they start to overlap generating a very rich dynamics. Tiny rounded borders (around 0.1% from the whole billiard size) are shown to be sufficient to generate the sticky motion with power-law decay γ(esc)=1.27, while borders larger than 10% are enough to produce escape times related to the chaotic motion. Escape exponents in the interval 1<γ(esc)<2 are generated due to marginal unstable periodic orbits trapping alternately (in time) regular and chaotic trajectories.