DISTRIBUTION OF PERIODIC-ORBITS AND THE HOMOCLINIC TANGLE

We study the distribution of regular and irregular periodic orbits on a Poincare surface of section of a simple Hamiltonian system of 2 degr ees of freedom. We explain the appearance of many lines of periodic or bits that form Farey trees. There are also lines thar are very close t o the asymptotic curves of the unstable periodic orbits. Some regular orbits, sometimes stable, are found inside the homoclinic tangle. We e xplain this phenomenon, which shows that the homoclinic tangle does no t cover the whole area around an unstable orbit, but has gaps. Inside the lobes only irregular orbits appear, and some of them are stable. W e conjecture that the opposite is also true, i.e. all irregular orbits are inside lobes.