Weak mixing and anomalous kinetics along filamented surfaces

We consider chaotic properties of a particle in a square billiard with a ho rizontal bar in the middle. Such a system can model field-line windings of the merged surfaces. The system has weak-mixing properties with zero Lyapun ov exponent and entropy, and it can be also interesting as an example of a system with intermediate chaotic properties, between the integrability and strong mixing. We show that the transport is anomalous and that its propert ies can be linked to the ergodic properties of continued fractions. The dis tribution of Poincare recurrences, distribution of the displacements, and t he moments of the truncated distribution of the displacements are obtained. Connections between different exponents are found. It is shown that the di stribution function of displacements and its truncated moments as a functio n of time exhibit log-periodic oscillations (modulations) with a universal period T-log=pi (2)/12 ln 2. We note that similar results are valid for a f amily of billiard, particularly for billiards with square-in-square geometr y. (C) 2001 American Institute of Physics.