# PUBLICATIONS

## I. Ergodic Billiards and their perturbations

2. #### Current in periodic Lorentz gases with twists. Commun. Math. Phys. 306, 747-776 (2011).

##### We study electrical current in two-dimensional periodic Lorentz gas in the presence of a twist force on the scatterers. In this deterministic system, billiard orbits are still geodesics between collisions, but do not reflect elastically when reaching the boundary. When the horizon is finite, i.e. the free flights between    collisions are bounded, the resulting current J is proportional to the strength of the twist force. We also prove the existence of a unique SRB measure, for which the Pesin entropy formula and Young's expression for the fractal dimension are valid.
3. Ergodicity of the generalized lemon billiards (with Jingyu Chen, Luke Mohr, Pengfei Zhang) Chaos, 2013, 23(4).
##### In this paper we study a two-parameter family of convex billiard tables, by taking the intersection of two round disks (with different radii) in the plane. These tables give a generalization of the one-parameter family of lemon-shaped billiards. Initially, there is only one ergodic table among all lemon tables. In our generalized family, we observe numerically the prevalence of ergodicity among the some perturbations of that table. Moreover, numerical estimates of the mixing rate of the billiard dynamics on some ergodic tables are also provided.
4. Free path for Lorentz gas with flat points. Continuous and Discrete Dynamical Systems,Vol 32  December (2012), 4445-4466.
##### In this paper we study a special family of Lorentz gas with infinite horizon. The periodic scatterers have C3 smooth boundary with  positive curvature except on finitely many flat points. In  addition there exists a trajectory with infinite free path and tangentially touching the scatterers only at some flat points. The discontinuity of the free path function is analyzed and we prove that the free path is piecewise Holder continuous with uniform Holder constant. In addition these systems are shown to be non-uniformly hyperbolic; local stable and unstable Manifolds exist on a set of full Lebesgue measure; and the stable and unstable holonomy maps are absolutely continuous.
5. Current for dispersing billiards under general forces (with Nikolai Chernov, Pengfei Zhang), Journal of Statistical Physics, 2013.
##### The Lorentz gas of Z2-periodic scatterers (or the so called Sinai bil- liards) can be used to model motion of electrons on an ionized medal. 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 sys- tem 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.
6. Stability and ergodicity of moon billiards (with Fatima Correia) Chaos 25, 083110 (2015)
##### 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.
7. On Another Edge of Defocusing: Hyperbolicity of Asymmetric Lemon Billiards, (with Bunimovich and P.Zhang), Communications in Mathematical Physics, 2016, Volume 341, Issue 3, pp 781–803.
##### It is well known that a way to construct chaotic (hyperbolic) billiards with focusing components is to place all regular components of the boundary of a billiard table sufficiently far away from the each focusing component. If all focusing components of the boundary of the billiard table are circular arcs, then the above condition states that all circles obtained by completion of focusing components to full circles belong to the billiard table. In the present paper we show that defocusing mechanism may generate hyperbolicity even in the case when this condition is strongly violated. We demonstrate that by proving that a class of convex tables--asymmetric lemons, whose boundary consists of two circular arcs, generate hyperbolic billiards. In such billiards the circle completing one of the boundary arcs contains the entire billiard table. Therefore this result is quite surprising because in our billiards, the focusing components are extremely close to each other, and because these tables are perturbations of the first convex ergodic billiard constructed more than forty years ago.
8. Ergodicity in umbrella billiards, (with Maria F Correia and Christopher Lee Cox), New Horizons in Mathematical Physics, 2017.

## II. Statistical Proeprties of Hyperbolic Systems

1. A family of chaotic billiards with variable mixing rates. Stochastics and Dynamics, (2005) 5, 535-553. (With N. Chernov).
##### 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/n^a (here n denotes the discrete time), in which the degree a in (1, \infty) changes continuously with the parameter of the family, p . We also derive an explicit relation between the degree a and the family parameter p.
2.     Billiards with Polynomial mixing rates. Nonlinearity, (2005) 18, 1527-1553.(With N. Chernov).
##### While many dynamical systems of mechanical origin, in particular billiards, are strongly chaotic and 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.
3.      Improved estimate for correlations in billiards , Communications in Mathematical Physics, 277 (2008), 305-321. (With N. Chernov)
4. #### On mixing rate for rectangular billiards with diamond obstacles, Am. J. Math. Manage. Sci. 30, No. 1-2, Spec. Issue, 53-65 (2012)

##### Billiards with focusing boundaries have nonuniform hyperbolicity and polynomial decay of correlations. Their dynamics exemplify a delicate transition from regular behavior to chaos, which makes them particularly interesting in mathematical physics. We present a rigorous analysis of correlations for a class of rectangular billiards with diamond obstacles, and prove the decay rate of correlations is of order 1/n.
5.   On statistical properties of hyperbolic systems with general singularities, (with Chernov) . Journal of Statistical Physics: Volume 136, Issue 4 (2009), Page 615-642
##### We study hyperbolic systems with singularities and prove the coupling lemma and exponential decay of correlations under weaker assumptions than previously adopted in similar studies. Our new approach allows us to study the mixing rates of the reduced map for certain billiard models that could not be handled by the traditional techniques. These models include modified Bunimovich stadia, which are bounded by minor arcs, and flower-type regions that are bounded by major arcs.
6.   Decay of Correlations on Non-Holder Observables,  International  Journal of Nonlinear Science, Vol. 10:3, 2010, 375--385.
##### This paper is devoted to investigating decay of correlations for hyperbolic systems with singularities on general continuous observables. We study the dependence of decay of correlations on the regularity of observables by the coupling method.
7. Estimates for correlations in billiards with large arcs. Acta Mathematicae Applicatae Sinica, English Series, Vol. 27, No. 3. (2011) 381-392.
9. #### A functional analytic approach to perturbations of the Lorentz gas (with Mark Demers), Communications in Mathematical Physics, 2013, 767-830.

##### 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.
11. Decay of correlations for billiards with flat points I: channel effect, Contemporary Mathematics, (2017).
##### In this paper we constructed a special family of semidispersing billiards bounded on a rectangle with a few dispersing scatters. We assume there exists a pair of flat points (with zero curvature) on the boundary of these scatters, whose tangent lines form a channel in the billiard table that is perpendicular to the vertical sides of the rectangle. The billiard can be induced to a Lorenz gas with infinite horizon when replacing the rectangle by a torus. We study the mixing rates of the one-parameter family of the semi-dispersing billiards and the Lorenz gas on a torus; and show that the correlation functions of both maps decay polynomially.
12. Decay of correlations for billiards with flat points II: cusp effect, Contemporary Mathematics, (2017).
##### In this paper we continue to study billiards with flat points, by constructing a spec family of dispersing billiards with cusps. All boundaries of the table have positive curvature except that the curvature vanishes at the vertex of a cusp, i.e. there is a pair of boundaries intersection at the flat point tangentially. We study the mixing rates of this one-parameter family of billiards with parameter $\beta\in (2,\infty)$, and show that the correlation functions of the collision map decay polynomially with order $\cO(n^{-\frac{1}{\beta-1}})$ as $n\to\infty$. In particular, this solves an open question raised by Chernov and Markarian \cite{CM05}
13. A Compound Poisson law for hitting times to periodic orbits in two-dimensional hyperbolic systems , (with Meagan Carney, Matthew Nicol) J. Statistical Physics, (2017).
##### We show that a compound Poisson distribution holds for scaled exceedances of observables φ uniquely maximized at a periodic point ζ in a variety of two-dimensional hyperbolic dynamical systems with singularities (M,T,μ), including the billiard maps of Sinai dispersing billiards in both the finite and infinite horizon case. The observable we consider is of form φ(z)=−lnd(z,ζ) where d is a metric defined in terms of the stable and unstable foliation. The compound Poisson process we obtain is a Pólya-Aeppli distibution of index θ. We calculate θ in terms of the derivative of the map T. Furthermore if we define Mn=max{φ,…,φ∘Tn} and un(τ) by limn→∞nμ(φ>un(τ))=τ the maximal process satisfies an extreme value law of form μ(Mn≤un)=e−θτ. These results generalize to a broader class of functions maximized at ζ, though the formulas regarding the parameters in the distribution need to be modified.
14. Stable laws for chaotic billiards with cusps at flat points, (with Paul Jung) Annales Henri Poincare, (2018).
##### We consider billiards with a single cusp where the walls meeting at the vertex of the cusp have zero one-sided curvature, thus forming a flat point at the vertex. For H\"older continuous observables, we show that properly normalized Birkoff sums, with respect to the billiard map, converge in law to a totally skewed $\alpha$-stable law.
15. Superdiffusions for certain nonuniformly hyperbolic systems, (with Luke Mohr) Submitted, (2018).
##### We investigate superdiffusion for stochastic processes generated by nonuniformly hyperbolic system models,in terms of the convergence of rescaled distributions to thenormal distribution following the abnormal centrallimit theorem, which differs from the usual requirement thatthe mean square displacement grow asymptoticallylinearly in time. We construct a martingale approximation that follows the idea of Doob’s decomposition theorem.We obtain an explicity formula for the superdiffusion constant in terms of the fine structure that originates inthe phase transitions as well as the geometry of the configuration domains of the systems. Models that satisfyour main assumptions include chaotic Lorentz gas, Bunimovich stadia, billiards with cusps, and can be apply toother nonuniformly hyperbolic systems with slow correlation decay rates of orderO(1/n)
16. Non-stationary Almost Sure Invariance Principle for Hyperbolic Systems with Singularities, , (with Yun Yang, Jianyu Chen) J. Statistical Phys, 2018, 1-26.
##### We investigate a wide class of two-dimensional uniformly hyperbolic systems with singularities, and prove the almost sure invariance principle (ASIP) for bounded dynamically Hölder observables, with a sharp rate O(n^{1/4+ε}) for any ε>0. Our results apply to Sinai dispersing billiards and their perturbations.
17. Fluctuation of the entropy production for the Lorentz gas under small external forces , (with Luc-Rey Bellet, Mark Demers), Communications in Mathematical Physics, (2018).
##### In this paper we study the physical and statistical properties of the periodicLorentz gas with finite horizon driven to a non-equilibrium steady state by the combi-nation of non-conservative external forces and deterministic thermostats. A version ofthis model was introduced by Chernov, Eyink, Lebowitz, and Sinai and subsequentlygeneralized by Chernov and the third author. Non-equilibrium steady states for thesemodels are SRB measures and they are characterized by the positivity of the steady stateentropy production rate. Our main result is to establish that the entropy production, in thiscontext equal to the phase space contraction, satisfies the Gallavotti–Cohen fluctuationrelation. The main tool needed in the proof is the family of anisotropic Banach spacesintroduced by the first and third authors to study the ergodic and statistical properties ofbilliards using transfer operator techniques
18. Central Limit Theorem for Billiards With Flat Points , (with Kien Nguyen), Springer Conference Proceedings Differential Equations and Dynamical Systems, 2018.
##### In this paper we study the physical and statistical properties of the periodicLorentz gas with finite horizon driven to a non-equilibrium steady state by the combi-nation of non-conservative external forces and deterministic thermostats. A version ofthis model was introduced by Chernov, Eyink, Lebowitz, and Sinai and subsequentlygeneralized by Chernov and the third author. Non-equilibrium steady states for thesemodels are SRB measures and they are characterized by the positivity of the steady stateentropy production rate. Our main result is to establish that the entropy production, in thiscontext equal to the phase space contraction, satisfies the Gallavotti–Cohen fluctuationrelation. The main tool needed in the proof is the family of anisotropic Banach spacesintroduced by the first and third authors to study the ergodic and statistical properties ofbilliards using transfer operator techniques
19. Improved Young Tower and Thermodynamic Formalism for Hyperbolic Systems with Singularities, (with Jianyu Chen, Fang Wang) submitted, (2018).
##### For the hyperbolic systems with singularities, Markov partitions are rather delicate to construct because of the fragmentation of the phase space by singularities. In this paper, we investigate the chaotic billiards and other related hyperbolic systems with singularities, and construct an improved Young tower whose first return to the base is Markov. This leads to a Markov partition of the phase space with countable states. %Our construction is based on the coupling lemma %for standard families. Stochastic properties with respect to the SRB measure immediately follow from our construction of the Markov partition, including the decay rates of correlations and the central limit theorem. We further establish the thermodynamic formalism for the family of geometric potentials, by using the inducing scheme of hyperbolic type on the tower base. All the results apply to Sinai dispersing billiards, and their small perturbations due to external forces and nonelastic reflections with kicks and slips.
20. Statistical Properties for 1-d Expanding Maps with Singularities of Low Regularity , (with Jianyu Chen, Yiwei Zhang) submitted, (2020).
##### We investigate the statistical properties of piecewise expanding maps on the unit interval, whose inverse Jacobian may have low regularity near singularities. By lifting the 1-d expanding map to a hyperbolic skew product of the unit square, we are able to apply the functional analytic method recently developed by Demers and Zhang under standard assumptions. We are able to prove that the 1-d expanding map admits an absolutely continuous, invariant SRB measure, with piecewise continuous density function. Furthermore, we establish the exponential decay of correlations, large deviation principle and the almost sure invariance principle for the 1-d map on a large class of observables. Our results apply to rather general 1-d expanding maps, including $C^1$ perturbations of the Lorenz-like map and the Gauss map, etc.
21. Convergence to a -stable Lévy motion for chaotic billiards with several cusps at flat points. , (with Paul Jung, Franccoise Pene), Nonlinearity. 2020.

## III. Random Billiards

1. The spectrum of the billiard Laplacian of a family of random billiards, (Joint with Renato Feres ) J. Statis. Phys. Vol. 141:6, 2010, 1039-1054.

## V Finanical Math & Networks

8. #### Kernels of certain differential operators, Nonlinear Funct. Anal. Appl. 12 (2007), 617-625. (with J. G. Lian)

9. Hyperbolic behavior of Jacobi fields on billiard flows , Impulsive and Hybrid Dynamical Systems, 2007, Waterloo, 1794-1798.(with J. G. Lian)