Positive Lyapunov Exponents Of Randomized Smooth Stadium Billiard

Billiard is a very popular research direction in dynamical systems.A lot of ques-tions,conjectures and research methods can be verified by constructing different types of billiards.For billiards satisfying some specific conditions,we study ergodicity,mixing,entropy,deacy of correlations and some other properties.An important assumption for the study of these properties is that the system is chaotic and even is required that the Lyapunov exponents have a positive lower bound.The research of chaotic billiards originated from the work of Sinai in 1970^[1],who proved that the diffusion type billiard is chaotic.For a long time this was the only known chaotic billiards.Untill around 1975,Sinai's student Bunimovich constructed a type of shrunken chaotic billiards which was later named after him.Stadium Billiard Q?L?is a very important class,but the boundary curve of Q?L?is not smooth at the points where the arc and the line are connected.After smoothing these points in a reasonable way,we get smooth Stadium Billiard Q_??L?.On the other hand,notice that the collision can be reduced when collision occurs on a straight line or on the same arc.We denote the reduced system of Q_??L?as?M⁺,F⁺?.Now we hope to prove that?M⁺,F⁺?is chaotic and estimate the lower bound of its Lyapunov exponents.Due to the switching of stable directions and unstable directions in the orbit of the particle,the estimation of the lower bound of the Lyapunov exponents for a deterministic system is of fundamental difficulty.In a recent article^[2],Jinxin Xue and his collaborators estimated the lower bound of Lyapunov exponents for some non-uniform hyperbolic systems including standard map by introducing randomness at each point of orbit.The introduction of randomness makes the estimation of lower bound of Lyapunov exponents be converted into the lower bound of a specific integral,through which we can avoid the difficulty of calculating limit.In this paper,we will apply the method developed by Xue et al.to Q_??L?.For the arc map F⁺obtained by reducing Q_??L?,we introduce reasonable randomness at each collision point.Then we would prove the existence of the Lyapunov exponents of the stochastic dynamical system?M⁺,F_?⁺?and estimate the lower bound of its Lyapunov exponents.