Abstract:
We present scanning-probe images and magnetic-field plots which reveal fractal conductance fluctuations in a quantum billiard. The quantum billiard is drawn and tuned using erasable electrostatic lithography, where the scanning probe draws patterns of surface charge in the same environment used for measurements. A periodicity in magnetic field, which is observed in both the images and plots, suggests the presence of classical orbits. Subsequent high-pass filtered high-resolution images resemble the predicted probability density of scarred wave functions, which describe the classical orbits.

Abstract:
Negatively biased surface-gates allow electrostatic depletion of selected regions of a 2DEG, forming confined regions of specific geometry called billiards, in which ballistic transport occurs. At millikelvin temperatures, the electron phase coherence length is sufficient that quantum interference effects produce reproducible magneto-conductance fluctuations (MCF) that act as a 'magneto-fingerprint' of the scattering dynamics in the billiard. It has been predicted that billiard MCF are fractal. Fractal MCF in mesoscopic semiconductor billiards are investigated experimentally. The MCF of a Sinai billiard displayed exact self-similarity (ESS). A correlation function analysis is used to quantify the presence of ESS. A model for the Sinai billiard MCF based on a Weierstrass function is presented. Using a bridging interconnect, a continuous transition between the Sinai and an empty square geometry is achieved. The removal of the circle induces a transition from ESS to statistical self-similarity (SSS), suggesting that ESS is due to the presence of an obstacle at the center of the billiard. The physical dependencies of SSS are investigated and show variation in the fractal dimension, rather than the fractal scaling range. SSS obeys a unified picture where the fractal dimension depends only on the ratio between the average spacing and broadening of billiard energy levels, irrespective of other billiard parameters. The semiclassical origin of SSS is demonstrated and the suppression of SSS is observed in both the quantum and classical limits. The influence of soft-wall potential profile on fractal MCF is investigated using double-2DEG billiards. Detailed reviews of semiconductor billiard fabrication, low-temperature electrical measurements and fractal analysis are also presented.

Abstract:
We consider the open stadium billiard, consisting of two semicircles joined by parallel straight sides with one hole situated somewhere on one of the sides. Due to the hyperbolic nature of the stadium billiard, the initial decay of trajectories, due to loss through the hole, appears exponential. However, some trajectories (bouncing ball orbits) persist and survive for long times and therefore form the main contribution to the survival probability function at long times. Using both numerical and analytical methods, we concur with previous studies that the long-time survival probability for a reasonably small hole drops like Constant/time; here we obtain an explicit expression for the Constant.

Abstract:
We prove that the Birkhoff sums for ``almost every'' relevant observable in the stadium billiard obey a non-standard limit law. More precisely, the usual central limit theorem holds for an observable if and only if its integral along a one-codimensional invariant set vanishes, otherwise a $\sqrt{n\log n}$ normalization is needed. As one of the two key steps in the argument, we obtain a limit theorem that holds in Young towers with exponential return time statistics in general, an abstract result that seems to be applicable to many other situations.

Abstract:
Diagram, known in theory of the Anderson localization as the Hikami box, is computed for the Sinai billiard. This interference effect is mostly important for trajectories tangent to the opening of the billiard. This diagram is universal at low frequencies, because of the particle number conservation law. An independent parameter, which we call phase volume of diffraction, determines the corresponding frequency range. This result suggests that level statistics of a generic chaotic system is not universal.

Abstract:
We present a systematic survey of scarring and symmetry effects in the stadium billiard. The localization of individual eigenfunctions in Husimi phase space is studied first, and it is demonstrated that on average there is more localization than can be accounted for on the basis of random-matrix theory, even after removal of bouncing-ball states and visible scars. A major point of the paper is that symmetry considerations, including parity and time-reversal symmetries, enter to influence the total amount of localization. The properties of the local density of states spectrum are also investigated, as a function of phase space location. Aside from the bouncing-ball region of phase space, excess localization of the spectrum is found on short periodic orbits and along certain symmetry-related lines; the origin of all these sources of localization is discussed quantitatively and comparison is made with analytical predictions. Scarring is observed to be present in all the energy ranges considered. In light of these results the excess localization in individual eigenstates is interpreted as being primarily due to symmetry effects; another source of excess localization, scarring by multiple unstable periodic orbits, is smaller by a factor of $\sqrt{\hbar}$.

Abstract:
We construct a well ordered symbolic dynamics plane for the stadium billiard. In this symbolic plane the forbidden and the allowed orbits are separated by a monotone pruning front, and allowed orbits can be systematically generated by sequences of approximate finite grammars.

Abstract:
A systematic numerical technique for the calculation of unstable periodic orbits in the stadium billiard is presented. All the periodic orbits up to order $p=11$ are calculated and then used to calculate the average Lyapunov exponent and the topological entropy. Applications to semiclassical quantization and to experiments in mesoscopic systems and microwave cavities are noted.

Abstract:
For the first time a three--dimensional (3D) chaotic billiard -- the 3D Sinai billiard -- was quantized, and high--precision spectra with thousands of eigenvalues were calculated. We present here a semiclassical and statistical analysis of the spectra, and point out some of the features which are genuine consequences of the three dimensionality of this chaotic billiard.

Abstract:
In the context of the semiclassical theory of short periodic orbits, scar functions play a crucial role. These wavefunctions live in the neighbourhood of the trajectories, resembling the hyperbolic structure of the phase space in their immediate vicinity. This property makes them extremely suitable for investigating chaotic eigenfunctions. On the other hand, for all practical purposes reductions to Poincare sections become essential. Here we give a detailed explanation of resonances and scar functions construction in the Bunimovich stadium billiard and the corresponding reduction to the boundary. Moreover, we develop a method that takes into account the departure of the unstable and stable manifolds from the linear regime. This new feature extends the validity of the expressions.