I talk about three interesting sub-themes bridging between nonlinear dynamics and quantum transport in mesoscopic billiards.
Firstly, triangular antidot lattices are investigated. We analyze the semiclassical conductivity of fully-chaotic triangular antidots in the low but intermediate magnetic field. Taking into account both a smooth classical part evaluated by the mean density of states and an oscillation part evaluated by periodic orbits, we find that the resistivity of the system yields a monotonic decrease with respect to the magnetic field. But when including the effect of orbit bifurcation due to the overlapping of a pair of periodic orbits, several distinguished peaks of resistivity appear. The theoretical results nicely explain both the locations and intensities of the anomalously large peaks observed in the experiment by NEC group (Phys. Rev. B51(1995)4649) .
Then, we shall proceed to investigation of open three-dimensional (3-d) quantum dots. Mixed phase-space structures of 3-d billiards show the Arnold diffusion that cannot be seen in 2-d billiards. A semiclassical conductance formula for ballistic 3-d billiards is derived. We find that, for partially- or completely-broken ergodic 3-d billiards such as SU(2) symmetric billiards, the dependence of the conductance on the Fermi wavenumber is dramatically changed by the lead orientation. As a symmetry-breaking weak magnetic field is applied, the conductance shows a tendency to grow. We conclude: In contrast to the 2-d case, the anomalous increment of the conductance should include a contribution arising from the (classical) Arnold diffusion as well as the (quantum) weak localization correction .
Finally, within a formalism of the semiclassical Kubo formula for conductivity, we give a periodic-orbits picture for the fractal magneto-conductance fluctuations recently observed in submicron-scale phase coherent ballistic billiards . The self-similar conductance fluctuations are shown to be caused by the self-similar unstable periodic orbits which are generated through a sequence of isochronous pitchfork bifurcations of straight-line orbits oscillating towards harmonic saddles. The saddles are universally created right at the point of contact with the leads or at certain places in the cavity as a consequence of the softwall confinement. Our mechanism is able to explain all the fractal-like magneto-conductance fluctuations in general softwall billiards .
Many other interesting themes in this field will be described in .
 J. Ma and K. Nakamura: Phys.Rev.B62 (2000) 13552-13556.
 J. Ma and K. Nakamura, cond-mat / 0108276 (2001).
 A. Budiyono and K. Nakamura, submitted for publication; see also a short communication by Dr. A. Budiyono in this Conference.
 K. Nakamura and T. Harayama, "Quantum Chaos and Quantum Dots" (Oxford University Press), to be published.