The hyperbolic Anosov C-systems have exponential instability of their trajectories and as such represent the most natural chaotic dynamical systems. Of special interest are C-systems which are defined on compact surfaces of the Lobachevsky plane of constant negative curvature. An example of such system has been introduced in a brilliant article published in 1924 by the mathematician Emil Artin. The dynamical system is defined on the fundamental region of the Lobachevsky plane which is obtained by the identification of points congruent with respect to the modular group, a discrete subgroup of the Lobachevsky plane isometries. The fundamental region in this case is a hyperbolic triangle. The geodesic trajectories of the non-Euclidean billiard are bounded to propagate on the fundamental hyperbolic triangle. In this article we shall expose his results, will calculate the correlation functions/observables which are defined on the phase space of the Artin billiard and demonstrate the exponential decay of the correlation functions with time. We use Artin symbolic dynamics, the differential geometry and group theoretical methods of Gelfand and Fomin.
We prove exponential decay of correlations for the billiard flow associated with a two-dimensional finite horizon Lorentz Gas (i.e., the Sinai billiard flow with finite horizon). Along the way, we describe the spectrum of the generator of the corresponding semi-group L_t of transfer operators, i.e., the resonances of the Sinai billiard flow, on a suitable Banach space of anisotropic distributions. (Revised after referees' comments.)
The open stadium billiard has a survival probability, $P(t)$, that depends on the rate of escape of particles through the leak. It is known that the decay of $P(t)$ is exponential early in time while for long times the decay follows a power law. In this work we investigate an open stadium billiard in which the leak is free to rotate around the boundary of the stadium at a constant velocity, $ω$. It is found that $P(t)$ is very sensitive to $ω$. For certain $ω$ values $P(t)$ is purely exponential while for other values the power law behaviour at long times persists. We identify three ranges of …
Chaos, namely exponential sensitivity to initial conditions, is generally considered a nuisance, inasmuch as it prevents long-term predictions in physical systems. Here, we present an easily accessible approach to undo deterministic chaos and tailor ray trajectories in arbitrary two-dimensional optical billiards, by introducing spatially varying refractive index therein. A new refractive index landscape is obtained by a conformal mapping, which makes the trajectories of the chaotic billiard fully predictable and the billiard fully integrable. Moreover, trajectory rectification can be pushed a …
We perform a detailed numerical study of diffusion in the $\varepsilon$ stadium of Bunimovich, and propose an empirical model of the local and global diffusion for various values of $\varepsilon$ with the following conclusions: (i) the diffusion is normal for all values of $\varepsilon \leq(0.3)$ and all initial conditions, (ii) the diffusion constant is a parabolic function of the momentum (i.e., we have inhomogeneous diffusion), (iii) the model describes the diffusion very well including the boundary effects, (iv) the approach to the asymptotic equilibrium steady state is exponential, (v) th…
