Osaka City University, Japan

Starting from evaluating a simple soft-wall multichannel billiard and using the semiclassical Kubo formula for conductivity, we gave a periodic-orbits picture for the exact self-similar magnetoconductance fluctuations obtained in the phase coherent ballistic quantum billiards experiments. The applicability of Kubo formula to these single devices are justified, since the dots are coupled to large source and drain so that due to level broadening, the energy level inside the dots can be considered as effectively continuum. In our picture the soft-wall boundary is experimentally essential. We claimed that the exact self-similar magnetoconductance fluctuations is due to the self-similar periodic orbits generated through a sequence of isochronous pitchfork bifurcations of straight-line liberating orbits oscillating toward harmonic saddles. The saddles are naturally created right at the point of contact with the leads or at certain places in the cavity as a consequence of the sofwall confinement. We will show that the function we obtained, approximately satisfies a scaling relation, which generates its self-similar properties. In fact, its spectrum can be approximated by the famous Weierstrass spectrum with an upper cut. In addition, we would like to show that the Hurst exponent of the fractal is independent of the detailed geometrical shapes of the cavity and determined only by the local information on the saddles. On this issue we would like to confirm the recent experimental findings that the fractal dimension of the fluctuations are determined only by the quantumness of the system. We choose to discuss the Hurst exponent which is defined as the ratio of the logarithmic values of the scaling constants in the y-direction and x-direction, instead of the fractal dimensions, since they are easy to calculate and to be confirmed experimentally for an exact self-affine fractal. Finally, using the same reasoning and again using the semiclassical Kubo formula for conductivity, we would like to show that even in the situation when the conductance fluctuations versus the applied weak magnetic field show fractal behaviour, the conductance fluctuations as a function of the Fermi energy are not fractal-like.