  The phase space of an area-preserving map typically contains infinitely many
elliptic islands embedded in a chaotic sea. Orbits near the boundary of a
chaotic region have been observed to stick for long times, strongly influencing
their transport properties. The boundary is composed of invariant "boundary
circles". We briefly report recent results of the distribution of rotation
numbers of boundary circles for the H\'enon quadratic map and show that the
probability of occurrence of small elements of their continued fraction
expansions is larger than would be expected for a number chosen at random.
However, large elements occur with probabilities distributed proportionally to
the random case. The probability distributions of ratios of fluxes through
island chains is reported as well. These island chains are neighbours in the
sense of the Meiss-Ott Markov-tree model. Two distinct universality families
are found. The distributions of the ratio between the flux and orbital period
are also presented. All of these results have implications for models of
transport in mixed phase space.
