  The Birkhoff Ergodic Theorem asserts under mild conditions that Birkhoff
averages (i.e. time averages computed along a trajectory) converge to the space
average. For sufficiently smooth systems, our small modification of numerical
Birkhoff averages significantly speeds the convergence rate for quasiperiodic
trajectories -- by a factor of $10^{25}$ for 30-digit precision arithmetic,
making it a useful computational tool for autonomous dynamical systems. Many
dynamical systems and especially Hamiltonian systems are a complex mix of
chaotic and quasiperiodic behaviors, and chaotic trajectories near
quasiperiodic points can have long near-quasiperiodic transients. Our method
can help determine which initial points are in a quasiperiodic set and which
are chaotic. We use our {\bf weighted Birkhoff average} to study quasiperiodic
systems, to distinguishing between chaos and quasiperiodicity, and for
computing rotation numbers for self-intersecting curves in the plane.
Furthermore we introduce the Embedding Continuation Method which is a
significantly simpler, general method for computing rotation numbers.
