Hong-Kun Zhang

Billiards with polynomial mixing rates

While many dynamical systems of mechanical origin, in particular billiards, are strongly chaotic—enjoy exponential mixing, the rates of mixing in many other models are slow (algebraic, or polynomial). The dynamics in the latter are intermittent between regular and chaotic, which makes them particularly interesting in physical studies. However, mathematical methods for the analysis of systems with slow mixing rates were developed just recently and are still difficult to apply to realistic models. Here, we reduce those methods to a practical scheme that allows us to obtain a nearly optimal bound on mixing rates. We demonstrate how the method works by applying it to several classes of chaotic billiards with slow mixing as well as discuss a few examples where the method, in its present form, fails.

A FAMILY OF CHAOTIC BILLIARDS WITH VARIABLE MIXING RATES

We describe a one-parameter family of dispersing (hence hyperbolic, ergodic and mixing) billiards where the correlation function of the collision map decays as 1/na (here n denotes the discrete time), in which the degree a ∈ (1, ∞) changes continuously with the parameter of the family, β. We also derive an explicit relation between the degree a and the family parameter β.

Dispersing Billiards with Moving Scatterers

We propose a model of Sinai billiards with moving scatterers, in which the locations and shapes of the scatterers may change by small amounts between collisions. Our main result is the exponential loss of memory of initial data at uniform rates, and our proof consists of a coupling argument for non-stationary compositions of maps similar to classical billiard maps. This can be seen as a prototypical result on the statistical properties of time-dependent dynamical systems.

A Functional Analytic Approach to Perturbations of the Lorentz Gas

We present a functional analytic framework based on the spectrum of the transfer operator to study billiard maps associated with perturbations of the periodic Lorentz gas. We show that recently constructed Banach spaces for the billiard map of the classical Lorentz gas are flexible enough to admit a wide variety of perturbations, including: movements and deformations of scatterers; billiards subject to external forces; nonelastic reflections with kicks and slips at the boundaries of the scatterers; and random perturbations comprised of these and possibly other classes of maps. The spectra and spectral projections of the transfer operators are shown to vary continuously with such perturbations so that the spectral gap enjoyed by the classical billiard persists and important limit theorems follow.

Spectral analysis of hyperbolic systems with singularities

We study the statistical properties of a general class of two-dimensional hyperbolic systems with singularities by constructing Banach spaces on which the associated transfer operators are quasi-compact. When the map is mixing, the transfer operator has a spectral gap and many related statistical properties follow, such as exponential decay of correlations, the central limit theorem, the identification of Ruelle resonances, large deviation estimates and an almost-sure invariance principle. To demonstrate the utility of this approach, we give two applications to specific systems: dispersing billiards with corner points and the reduced maps for certain billiards with focusing boundaries.

Regularity of Bunimovich’s stadia

Stadia are popular models of chaotic billiards introduced by Bunimovich in 1974. They are analogous to dispersing billiards due to Sinai, but their fundamental technical characteristics are quite different. Recently many new results were obtained for various chaotic billiards, including sharp bounds on correlations and probabilistic limit theorems, and these results require new, more powerful technical apparatus. We present that apparatus here, in the context of stadia, and prove “regularity” properties.

Electrical Current in Sinai Billiards Under General Small Forces

The Lorentz gas of

Stability and ergodicity of moon billiards.

\mathbb{Z}^{2}

Multiple scattering in random mechanical systems and diffusion approximation

-periodic scatterers (or the so called Sinai billiards) can be used to model motion of electrons on a metal. We investigate the linear response for the system under various external forces (during both the flight and the collision). We give some characterizations under which the forced system is time-reversible, and derive an estimate of the electrical current generated by the forced system. Moreover, applying Pesin entropy formula and Young dimension formula, we get several characterizations of the non-equilibrium steady state of the forced system.

Ergodicity of the generalized lemon billiards

We construct a two-parameter family of moon-shaped billiard tables with boundary made of two circular arcs. These tables fail the defocusing mechanism and other known mechanisms that guarantee ergodicity and hyperbolicity. We analytically study the stability of some periodic orbits and prove there is a class of billiards in this family with elliptic periodic orbits. These moon billiards can be viewed as generalization of annular billiards, which all have Kolmogorov-Arnold-Moser islands. However, the novelty of this paper is that by varying the parameters, we numerically observe a subclass of moon-shaped billiards with a single ergodic component.