Mixing property of symmetrical polygonal billiards.

The present work consists of a numerical study of the dynamics of irrational polygonal billiards. Our contribution reinforces the hypothesis that these systems can be strongly mixing, although never demonstrably chaotic, and discusses the role of rotational symmetries on the billiards boundaries. We introduce a biparametric polygonal billiard family with only C_{n} rotational symmetries. Initially, we calculate through the relative measure r(ℓ,θ;t) the phase space filling. This is done for some integer values of n and for a plane of parameters ℓ×θ. From the resulting phase diagram, we can identify the fully ergodic systems. The numerical evidence that symmetrical polygonal billiards can be strongly mixing is obtained by evaluating the position autocorrelation function Cor_{x}(t), which follows a power-law-type decay t^{-σ}. The strongly mixing property is indicated by σ=1. For odd, small values of n, the exponent σ≃1 is found. On the other hand, σ<1 (weakly mixing cases) for small, even values of n. Intermediate n values present σ≃1 independently of parity. For larger values of symmetry parameter n, the biparametric family tends to be a circular billiard (integrable case). For such values of n, we identified even less ergodic behavior at the pace at which n increases and σ decreases.

Time-reversal-invariant hexagonal billiards with a point symmetry.

A biparametric family of hexagonal billiards enjoying the C_{3} point symmetry is introduced and numerically investigated. First, the relative measure r(ℓ,θ;t) in a reduced phase space was mapped onto the parameter plane ℓ×θ for discrete time t up to 10^{8} and averaged in tens of randomly chosen initial conditions in each billiard. The resulting phase diagram allowed us to identify fully ergodic systems in the set. It is then shown that the absolute value of the position autocorrelation function decays like |C_{q}(t)|∼t^{-σ}, with 0<σ⩽1 in the hexagons. Following previous examples of irrational triangles, we were able to find billiards for which σ∼1. This is further evidence that, although not chaotic (all Lyapunov exponents are zero), billiards in polygons might exhibit a near strongly mixing dynamics in the ergodic hierarchy. Quantized counterparts with distinct classical properties were also characterized. Spectral properties of singlets and doublets of the quantum billiards were investigated separately well beyond the ground state. As a rule of thumb, for both singlet and doublet sequences, we calculate the first 120 000 energy eigenvalues in a given billiard and compute the nearest neighbor spacing distribution p(s), as well as the cumulative spacing function I(s)=∫_{0}^{s}p(s^{'})ds^{'}, by considering the last 20 000 eigenvalues only. For billiards with σ∼1, we observe the results predicted for chaotic geometries by Leyvraz, Schmit, and Seligman, namely, a Gaussian unitary ensemble behavior in the degenerate subspectrum, in spite of the presence of time-reversal invariance, and a Gaussian orthogonal ensemble behavior in the singlets subset. For 0<σ<1, formulas for intermediate quantum statistics have been derived for the doublets following previous works by Brody, Berry and Robnik, and Bastistic and Robnik. Different regimes in a given energy spectrum have been identified through the so-called ergodic parameter α=t_{H}/t_{C}, the ratio between the Heisenberg time and the classical diffusive-like transport time, which signals the possibility of quantum dynamical localization when α<1. A good quantitative agreement is found between the appropriate formulas with parameters extracted from the classical phase space and the data from the calculated quantum spectra. A rich variety of standing wave patterns and corresponding Poincaré-Husimi representations in a reduced phase space are reported, including those associated with lattice modes, scarring, and high-frequency localization phenomena.