PUBLICATIONS
I. Ergodic Billiards and their perturbations

Regularity of Bunimovich's stadia , Regular and Chaotic Dynamics, 3 (2007) , 335356 . (With N. Chernov)
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.

We study electrical current in twodimensional 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.
 Ergodicity of the generalized lemon billiards (with Jingyu Chen, Luke Mohr, Pengfei Zhang) Chaos, 2013, 23(4).
In this paper we study a twoparameter 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 oneparameter family of lemonshaped 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.
 Free path for Lorentz gas with flat points. Continuous and Discrete Dynamical Systems,Vol 32 December (2012), 44454466.
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 nonuniformly hyperbolic; local stable and unstable Manifolds exist on a set of full Lebesgue measure; and the stable and unstable holonomy maps are absolutely continuous.
 Current for dispersing billiards under general forces (with Nikolai Chernov, Pengfei Zhang), Journal of Statistical Physics, 2013.
The Lorentz gas of Z2periodic 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 timereversible, 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 nonequilibrium steady state of the forced system.
 Stability and ergodicity of moon billiards (with Fatima Correia) Chaos 25, 083110 (2015)
We construct a twoparameter family of moonshaped 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 KolmogorovArnoldMoser islands. However, the novelty of this paper is that by varying the parameters, we numerically observe a subclass of moonshaped billiards with a single ergodic component.
 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 tablesasymmetric 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.
 Ergodicity in umbrella billiards, (with Maria F Correia and Christopher Lee Cox), New Horizons in Mathematical Physics, 2017.
We investigate a threeparameter family of billiard tables with circular arc boundaries. These umbrellashaped billiards may be viewed as a generalization of twoparameter moon and asymmetric lemon billiards, in which the latter classes comprise instances where the new parameter is $0$. Like those two previously studied classes, for certain parameters umbrella billiards exhibit evidence of chaotic behavior despite failing to meet certain criteria for defocusing or dispersing, the two most well understood mechanisms for generating ergodicity and hyperbolicity. For some parameters corresponding to nonergodic lemon and moon billiards, small increases in the new parameter transform elliptic $2$periodic points into a cascade of higher order elliptic points. These may either stabilize or dissipate as the new parameter is increased. We characterize the periodic points and present evidence of new ergodic examples.
II. Statistical Proeprties of Hyperbolic Systems
 A family of chaotic billiards with variable mixing rates. Stochastics and Dynamics, (2005) 5, 535553. (With N. Chernov).
We describe a oneparameter 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.
 Billiards with Polynomial mixing rates. Nonlinearity, (2005) 18, 15271553.(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.
 Improved estimate for correlations in billiards , Communications in Mathematical Physics, 277 (2008), 305321. (With N. Chernov)
We consider several classes of chaotic billiards with slow (polynomial) mixing rates, which include Bunimovich stadium and dispersing billiards with cusps. In recent papers by Markarian and the present authors, estimates on the decay of correlations were obtained that were suboptimal (they contained a redundant logarithmic factor). We sharpen those estimates by removing that factor.

On mixing rate for rectangular billiards with diamond obstacles, Am. J. Math. Manage. Sci. 30, No. 12, Spec. Issue, 5365 (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.
 On statistical properties of hyperbolic systems with general singularities, (with Chernov) . Journal of Statistical Physics: Volume 136, Issue 4 (2009), Page 615642
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 flowertype regions that are bounded by major arcs.
 Decay of Correlations on NonHolder Observables, International Journal of Nonlinear Science, Vol. 10:3, 2010, 375385.
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.
 Estimates for correlations in billiards with large arcs. Acta Mathematicae Applicatae Sinica, English Series, Vol. 27, No. 3. (2011) 381392.
Bunimovich billiards areergodic and mixing. However, if the billiard table contains very large arcs on its boundary, then there exist trajectories experience in finitely many collisions in the vicinity of periodic trajectories on the large arc. The hyperbolicity is nonuniform and mixing rate is very slow. The corresponding dynamics are intermittent between regular and chaotic, which makes them particularly interesting in physical studies. The study of mixing rates in intermittent chaotic systems is more difficult than that of truly chaotic ones, and the resulting estimates may depend on delicate details of the dynamics in the traps. We present a rigorous analysis of the corresponding singularities and correlations to certain class of billiards and show the rate of mixing is of order 1/n.

We study the billiard map associated with both the finite and infinite horizon Lorentz gases having smooth scatterers with strictly positive curvature. We introduce generalized function spaces (Banach spaces of distributions) on which the transfer operator is quasicompact. The mixing properties of the billiard map then imply the existence of a spectral gap and related statistical properties such as exponential decay of correlations and the central limit theorem. Finer statistical properties of the map such as the identification of Ruelle resonances, large deviation estimates and an almostsure invariance principle follow immediately once the spectral picture is established.

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 exible enough to admit a wide variety of perturbations, including: movements and deformations of scatterers; billiards subject to external forces; nonelastic re ections 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.

We study the statistical properties of a general class of twodimensional hyperbolic systems with singularities by constructing Banach spaces on which the associated transfer operators are quasicompact. 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 almostsure 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.
 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 oneparameter family of the semidispersing billiards and the Lorenz gas on a torus; and show that the correlation functions of both maps decay polynomially.
 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 oneparameter family of billiards with parameter $\beta\in (2,\infty)$, and show that the correlation functions of the collision map decay polynomially with order $\cO(n^{\frac{1}{\beta1}})$ as $n\to\infty$. In particular, this solves an open question raised by Chernov and Markarian \cite{CM05}
 A Compound Poisson law for hitting times to periodic orbits in twodimensional 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 twodimensional 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 PólyaAeppli 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.
 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 onesided 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.
 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)
 Nonstationary Almost Sure Invariance Principle for Hyperbolic Systems with Singularities,
, (with Yun Yang, Jianyu Chen) J. Statistical Phys, 2018, 126.
We investigate a wide class of twodimensional 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.
 Fluctuation of the entropy production for the Lorentz gas under small external forces
, (with LucRey 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 nonequilibrium steady state by the combination of nonconservative 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. Nonequilibrium 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
 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 nonequilibrium steady state by the combination of nonconservative 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. Nonequilibrium 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
 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.
 Statistical Properties for 1d 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 1d 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 1d 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 1d map on a large class of observables.
Our results apply to rather general 1d expanding maps, including $C^1$ perturbations of
the Lorenzlike map and the Gauss map, etc.
 Convergence to a stable Lévy motion for chaotic billiards with several cusps at flat points.
, (with Paul Jung, Franccoise Pene), Nonlinearity. 2020.
We consider billiards with several possibly nonisometricand asymmetric cusps at flat points; the caseof a single symmetric cusp was studied previously in [Zha17]and [JZ18]. In particular, we show thatproperly normalized Birkhoff sums of Hölder observables, with respect to the billiard map, convergein Skorokhod’sM1topology to anαstable Lévy motion, whereαdepends on the ‘curvature’ of theflattest points and the skewness parameterξdepends on the values of the observable at those same points.Previously, [JZ18] proved convergence of the onepoint marginals to totally skewedαstable distributionsfor a symmetric cusp. The limits we prove here are stronger, since they are in the functional sense, but alsoallow for more varied behaviour due to the presence of multiple cusps. In particular, the general limits weobtain allow for any skewness parameter, as opposed to just the totally skewed cases. We also show thatconvergence in the strongerJ1topology is not possible.
III. Random Billiards
 The spectrum of the billiard Laplacian of a family of random billiards, (Joint with Renato Feres ) J. Statis. Phys. Vol. 141:6, 2010, 10391054.
Random billiards are billiard dynamical systems for which the reflection law giving the postcollision direction of a billiard particle as a function of the precollision direction is specified by a Markov (scattering) operator P . Billiards with microstructure are random billiards whose Markov operator is derived from a ``microscopic surface structure" on the boundary of the billiard table. The microstructure in turn is defined in terms of what we call a billiard cell Q, the shape of which completely determines the operator P . This operator, defined on an appropriate Hilbert space, is bounded selfadjoint and, for the examples considered here, a HilbertSchmidt operator. A central problem in the statistical theory of such random billiards is to relate the geometric characteristics of Q and the spectrum of P . We show, for a particular family of billiard cell shapes parametrized by a scale invariant curvature K (Fig. 2), that the billiard Laplacian P −I is closely related to the ordinary spherical Laplacian, and indicate, by partly analytical and partly numerical means, how this provides asymptotic information about the spectrum of P for small values of K. It is shown, in particular, that the second moment of scattering about the incidence angle closely approximates the spectral gap of P .

A random billiard is a random dynamical system similar to an ordinary billiard system except that the standard specular reflection law is replaced with a more general stochastic operator specifying the postcollision distribution of velocities for any given precollision velocity. We consider such collision operators for certain random billiards we call billiards with microstructure. Collisions modeled by these operators can still be thought of as elastic and time reversible. The operators are canonically determined by a second (deterministic) billiard system that models ``microscopic roughness" on the billiard table boundary. Our main purpose here is to develop some general tools for the analysis of the collision operator of such random billiards. Among the main results, we give geometric conditions for these operators to be HilbertSchmidt and relate their spectrum and speed of convergence to stationary Markov chains with geometric properties of the microscopic billiard structure. The relationship between spectral gap and the shape of the microstructure is illustrated with several simple examples.

diffusion approximation (with Renato Feres and Jasmine Ng). Commun. Math. Phys. 323, 713  745 (2013).
This paper is concerned with stochastic processes that model multiple (or iterated) scattering in classical mechanical systems of billiard type, defined below. From a given (deterministic) system of billiard type, a random process with transition probabilities operator P is introduced by assuming that some of the dynamical variables are random with prescribed probability distributions. Of particular interest are systems with weak scattering, which are associated to parametric families of operators P_h, depending on a geometric or mechanical parameter h, that approaches the identity as h goes to 0. It is shown that (P_h I)/h converges for small h to a second order elliptic differential operator L on compactly supported functions and that the Markov chain process associated to P_h converges to a diffusion with infinitesimal generator L. Both P_h and L are selfadjoint (densely) defined on the space L2(H,{\eta}) of squareintegrable functions over the (lower) halfspace H in R^m, where {\eta} is a stationary measure. This measure's density is either (postcollision) MaxwellBoltzmann distribution or Knudsen cosine law, and the random processes with infinitesimal generator L respectively correspond to what we call MB diffusion and (generalized) Legendre diffusion. Concrete examples of simple mechanical systems are given and illustrated by numerically simulating the random processes.

We consider random flights of point particles inside $n$dimensional channels of the form $\mathbb{R}^{k} \times \mathbb{B}^{nk}$, where $\mathbb{B}^{nk}$ is a ball of radius $r$ in dimension $nk$. The particle velocities immediately after each collision with the boundary of the channel comprise a Markov chain with a transition probabilities operator $P$ that is determined by a choice of (billiardlike) random mechanical model of the particlesurface interaction at the "microscopic" scale. Our central concern is the relationship between the scattering properties encoded in $P$ and the constant of diffusivity of a Brownian motion obtained by an appropriate limit of the random flight in the channel. Markov operators obtained in this way are {\em natural} (definition below), which means, in particular, that (1) the (at the surface) MaxwellBoltzmann velocity distribution with a given surface temperature, when the surface model contains moving parts, or (2) the socalled Knudsen cosine law, when this model is purely geometric, is the stationary distribution of $P$. We show by a suitable generalization of a central limit theorem of Kipnis and Varadhan how the diffusivity is expressed in terms of the spectrum of $P$ and compute, in the case of 2dimensional channels, the exact values of the diffusivity for a class of parametric microscopic surface models of the above geometric type (2).

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. Ourmain result is the exponential loss ofmemory of initial data at uniform rates, and our proof consists of a coupling argument for nonstationary compositions of maps similar to classical billiard maps. This can be seen as a prototypical result on the statistical properties of timedependent dynamical systems.

Rigid bodies collision maps in dimension two, under a natural set of physical requirements, can be classified into two types: the standard specular reflection map and a second which we call, after Broomhead and Gutkin, noslip. This leads to the study of noslip billiardsplanar billiard systems in which the moving particle is a disc (with rotationally symmetric mass distribution) whose translational and rotational velocities can both change at each collision with the boundary of the billiard domain. In this paper we greatly extend previous results on boundedness of orbits (Broomhead and Gutkin) and linear stability of periodic orbits for a Sinaitype billiard (Wojtkowski) for noslip billiards. We show among other facts that: (i) for billiard domains in the plane having piecewise smooth boundary and at least one corner of inner angle less than $\pi$, noslip billiard dynamics will always contain elliptic period$2$ orbits; (ii) polygonal noslip billiards always admit small invariant open sets and thus cannot be ergodic with respect to the canonical invariant billiard measure; (iii) the noslip version of a Sinai billiard must contain linearly stable periodic orbits of period $2$ and, more generally, we provide a curvature threshold at which a commonly occurring period$2$ orbit shifts from being hyperbolic to being elliptic; (iv) finally, we make a number of observations concerning periodic orbits in a class of polygonal billiards.

We study limit theorems in the context of random perturbations of dispersing billiardsin finite and infinite measure. In the context of a planar periodic Lorentz gas with finite horizon,we consider random perturbations in the form of movements and deformations of scatterers. Weprove a Central Limit Theorem for the cell index of planar motion, as well as a mixing Local LimitTheorem for the cell index with piecewise Hölder continuousobservables. In the context of theinfinite measure random system, we prove limit theorems regarding visits to new obstacles andselfintersections, as well as decorrelation estimates. The main tool we use is the adaptation ofanisotropic Banach spaces to the random setting.
IV Quantum Billiards

In the paper, we establish the dynamical spectral rigidity for piecewise analyticBunimovich stadia and squashtype stadia. In addition, for smooth Bunimovichstadia and squashtype stadia we compute the Lyapunov eigenvalues along themaximal period two orbit, as well as value of the Peierls’ Barrier function from themarked length spectrum associated to the rotation numbern/(2n+ 1).

On Quantum unique ergodicity for Sinai billiards, (with Connor Kennedy). In preparation.
V Finanical Math & Networks

Evaluation of the Stochastic Modeling on Options, (with Zhijuan Mao, Zhian Liang, Jinguo Lian) International Journal of Engineering Research and Applications, 2 (2012): .24632473.
Modern option pricing techniques are often considered among the most mathematically complex of all applied areas of financial engineering. In particular these techniques derive their impetus from four milestones of option pricing models: Bachelier model, Samuelson model, Black ScholesMerton model and Levy model. In this paper we evaluate all related option pricing models based on these milestones, by comparing the corresponding stochastic differential equations and option pricing formulas. In addition we also include some simulations to make the comparisons more transparent.

The data analysis of the metal markets has recently attracted a lot of attention, mainly because the prices of precious metal are relatively more volatile than its historical trend. A robust estimate of extreme loss is vital, especially for mining companies to mitigate risk and uncertainty in metal price fluctuations. This paper examines the ValueatRisk and statistical properties in daily price return of precious metals, which include gold, silver, platinum, and palladium, from January 11, 2000 to September 9, 2016. An advanced two stage approach which combining GARCHtype models with Extreme Value Theory is implemented. In the first stage, the conditional variance is modeled by different rolling univariate GARCHtype models (GARCH, EGARCH and TGARCH) under the GED error assumption in the returns of precious metal markets and compare the same with other wellknown models. In the second stage, Extreme Value approach is applied to capture the tail behavior of distribution for the extracted standardized residuals. In comparison with the dynamic VaRs of these precious metals, we find that gold has the most steady and the highest VaRs, followed by platinum and silver; on the other hand our results show that palladium has the most volatile VaRs. The backtesting result confirms that our approach is an adequate method in improving risk management assessments and hedging strategies in the high volatile metal markets. 3.

Calendar Effects in AAPL ValueatRisk, (with Zhijing Zhang) Journal of Mathematics and System Science 6(6) · May 2016.
This study investigates calendar anomalies: dayoftheweek effect and seasonal effect in the ValueatRisk (VaR) analysis of stock returns for AAPL during the period of 1995 through 2015. The statistical properties are examined and a comprehensive set of diagnostic checks are made on the two decades of AAPL daily stock returns. Combing the Extreme Value Approach together with a statistical analysis , it is learnt that the lowest VaR occurs on Fridays and Mondays typically. Moreover, high Q4 and Q3 VaR are observed during the test period. These results are valuable for anyone who needs evaluation and forecasts of the risk situation in AAPL. Moreover, this methodology, which is applicable to any other stocks or portfolios, is more realistic and comprehensive than the standard normal distribution based VaR model that is commonly used.

Predicting the Risk of Bankruptcy for ARO Stock, (with Yuexian Li, Jinguo Lian); International Journal of Engineering Research & Science; (2015) 1:9.
In the last several decades, many researchers have been focused on finding effective experimental methods to predict stocks with tendency of bankrupt. Recent financial crisis has caused extensive worldwide economic damages, predicting bankruptcy before it happens could help investors avoid large losses. In this article, by observing the market dynamics of its stock price and trading volume, we estimate the risk of bankruptcy of Aeropostale (ARO). GARCH and EGARCH series models with normal distribution and tstudent distribution are used to estimate the volatilities and valueatrisk (VaR) of ARO stock. By analyzing the VaR, we conclude that there is a high probability that the company will be facing bankruptcy in the near future. Moreover, our study shows that the asymmetric EGARCH model with tstudent distribution eventually is a better choice to predict the behavior of this stock.

The oil price has a very important affection to the world economy. In this paper, using data sets of Europe Brent and West Texas Intermediate (WTI) Cushing crude oil daily prices from Jan. 4, 2000 to Jan. 4, 2016, the VaR forecasting performance of GARCHtype models are analyzed and compared in a short horizon; based on the Kupiecs POFtest and Christoffersens interval forecast test, as well as a Backtesting VaR Loss Function, The empirical results indicate that, for Europe Brent crude oil, EGARCH(1,1) has the best performance; while for WTI, APARCH(1,1) and GJRGARCH(1,1) outperform other GARCH models. In fact, these results also have significant guidance about how to choose a better Risk management model for the certain commodity of different companies even in the same time period.

A COPULAGARCH Approach of conditional dependence among oil, gold and US Dollar exchange rates. (with Zheng Wei, Zijing Zhang), HELIYON (2018)

In this paper, we study a class of differential operators which are second order, matrix coefficient Schrodinger operators with infinitely many jump conditions. Our operators are more general than those occurring in dispersive billiards, in other words, dispersive geodesic flow on Riemannian manifold with boundary, but include these as a special case. In this case, the jump conditions correspond to the reflections. A derivation of the jump conditions for dispersive billiards on m dimensional Riemannian manifold with boundary is included.

Kernels of certain differential operators, Nonlinear Funct. Anal. Appl. 12 (2007), 617625. (with J. G. Lian)
 Hyperbolic behavior of Jacobi fields on billiard flows , Impulsive and Hybrid Dynamical Systems, 2007, Waterloo, 17941798.(with J. G. Lian)
This paper discusses hyperbolic behavior of Jacobi fields along billiard flows on multidimensional Remannian manifolds. A Class of generalized differential operators associated with the impulsive equations (generalized Jacobi equations) are defined using a new Radon measure. We investigate the hyperbolic behavior of functions in the nullspace of the operator by applying operator theory.

Stability of the Tperiodic Solutions on the ESS Model . Rocky Mountain Journal of Mathematics, 38, (2008), 14931504. (With J.G. Lian)
In this paper, by employing the powerful and effective coincidence degree method, we show the existence of Tperiodic solutions of the extended simplified Schnakeberg (ESS) model in a strictly positively invariant region. Furthermore, Floquet theory if provided to show that the Tperiodic solution is unique and locallyasymptotically stable.

Unique periodic solution of ESS model, Appl. Math. ENotes, 8 (2008), 2529 (with J. G. Lian)

This paper provides a new approach to study the solutions of a class of generalized Jacobi equations associated with the linearization of certain singular ows on Riemannian manifolds with dimension n + 1. A new class of generalized differential operators is defined. We investigate the kernel of the corresponding maximal operators by applying operator theory. It is shown that all nontrivial solutions to the generalized Jacobi equation are hyperbolic, in which there are n dimension solutions with exponentialdecaying amplitude.