1. Coupling lemma for unbounded observables.
    (with Fang Wang and Hong-Kun Zhang)
    Main result: In this article, we study the decay rates of the correlation functions for a hyperbolic system \(T:M\to M\) with singularities that preserves a unique mixing SRB measure \(\mu\). We prove that, under some general assumptions, the correlations \(C_n(f,g)\) decay exponentially as \(n\to \infty\) for each pair of piecewise Holder observables \(f, g \in L_p(\mu)\) and for each \(p > 1\). As an application, we prove that the autocorrelations of the first return time functions decay exponentially for the induced maps of various billiard systems, which include the semi-dispersing billiards on a rectangle, billiards with cusps, and Bunimovich stadia (for the truncated first return time functions). These estimates of the decay rates of autocorrelations of the first return time functions for the induced maps have an essential importance in the study of the statistical properties of nonuniformly hyperbolic systems (with singularities).

  2. Hyperbolicity of asymmetric lemon billiards.
    (with Xin Jin)
    Main result: Asymmetric lemon billiards was introduced in [CMZZ], where the billiard table \(Q(r,b,R)\) is the intersection of two round disks with radii \(r\le R\), respectively, and \(b\) measures the distance between the two centers. The boundary consists of two circular arcs \(\Gamma_r\) and \(\Gamma_R\). It is conjectured [BZZ] that the asymmetric lemon billiards is hyperbolic when the arc \(\Gamma_r\) is a major arc and \(\Gamma_R\) is large. In this paper we prove this conjecture for sufficiently large \(R\).

  3. Twist interval for twist maps.
    Main result: The twist interval of a twist map on the annulus \(A=T \times [0,1]\) has nonempty interior if \(f\) preserves the area, which could be degenerate for general twist maps. In this note, we show that if a twist map \(f\) is non-wandering, then the twist interval of \(f\) is non-degenerate. Moreover, if there are two disjoint invariant curves of \(f\), then their rotation numbers must be different (no matter whether they are rational or irrational).

  4. Homoclinic intersections of symplectic partially hyperbolic systems with 2D center.
    Main result: We prove that there is an open class of symplectic diffeomorphisms among those partially hyperbolic systems with 2D center, such that for a \(C^r\)-generic map in this set, any hyperbolic periodic point admits some transverse homoclinic intersection.

  5. Papers

  6. Homoclinic intersections for geodesic flows on convex spheres.
    (with Zhihong Xia)
    Contemporary Math. 698 (2017), 221--238. arXiv
    Main result: Some geometrical properties of Riemannian manifolds can be characterized by the dynamical properties of the geodesic flows induced by the Riemannian metric. The Bumpy Metric Theorem states that generically, every closed geodesic is either hyperbolic or irrationally elliptic. In this paper we study some generic properties of the geodesic flows on a convex sphere. We prove that, \(C^r\) generically (\(2\le r\le \infty\)), every hyperbolic closed geodesic admits some transverse homoclinic intersections.

  7. Convex billiards on convex spheres.
    Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), 793--816. DOI; arXiv.
    Main result: We consider the dynamical billiards on convex domains on convex spheres. We prove that \(C^r\)-generically, every periodic point is either hyperbolic or elliptic with irrational rotation number (Kupka-Smale). Moreover, there exist transverse homoclinic intersections for every hyperbolic periodic point (Poincare's connecting problem). In particular, open and densely, convex billiards on a convex sphere have positive topological entropy.

  8. Accessibility and homology bounded strong unstable foliation for Anosov diffeomorphisms on 3-torus.
    (with Yan Ren and Shaobo Gan)
    Acta Math. Sinica 33 (2017), 71--76. DOI; arXiv.
    Main result: A hyperbolic matrix \(A \in \text{SL}(3,\mathbb{Z})\) induces a (linear) action on \(\mathbb{T}^3\), which can be viewed either as a uniformly hyperbolic map, or as a partically hyperbolic map with a 1D center. In this paper we the nonlinear ones, that is, Anosov diffeomorphisms \(f\) on \(\mathbb{T}^3\). We showed that the subbundle \(E^u_f \oplus E^s_f\) is integrable if and only if the strong unstable leaves of \(f\) lie in a uniform neighborhood of the ones of the linear model. This gives a negative answer to Hammerlindl's question about homology boundedness of strong unstable foliation.

  9. On another edge of defocusing: hyperbolicity of asymmetric lemon billiards.
    (with Leonid Bunimovich and Hong-Kun Zhang)
    Comm. Math. Phys. 341 (2016), 781--803. DOI; arXiv.
    Main result: There are two known mechanisms that generate hyperbolic billiards: the dispersing mechanism discovered by Sinai in 1970, and the defocusing mechanism discovered by Bunimovich in 1974. In this paper we study a new family of billiards: asymmetric lemons \(Q(r,b,R)\), obtained as the intersection of two disks. Note that none of the tables in this family satisfies the separation condition of the defocusing mechanism. We proved that the billiard on \(Q(r,b,R)\) is completely hyperbolic for an open set of parameters. Our study depends heavily on the manipulations of finite continued fractions.

  10. Homoclinic points for convex billiards.
    (with Zhihong Xia)
    Nonlinearity 27 (2014), 1181--1192. DOI; arXiv.
    Main result: The only way to perturb a billiard system is to deform the billiard table. Such deformations always produce non-local perturbations of the billiard dynamics, which make the study of \(C^r\)-generic properties of billiard systems a difficult task. We prove that generically, every hyperbolic periodic point of a convex billiard system has some transverse homoclinic intersections. This solve partially Poincare's connecting problem for billiard systems. Note that there are infinitely many periodic points for any convex billiards. So our result implies that open and dense convex billiards have positive topological entropy.

  11. Fundamental domain of invariant sets and applications.
    Ergod. Th. Dynam. Syst. 34 (2014), no. 1, 341--352. DOI; arXiv.
    Main result: Let \(E\) be an invariant subset of a dynamical system. A dynamical fundamental domain is a subset of \(E)\) that intersects each orbit at exactly one points. We prove the existence of fundamental domains and gave several applications to hyperbolic and partially hyperbolic systems. In particular, we obtain a dichotomy for an accessible partially hyperbolic systems, that either it is completely dissipative, or it must be ergodic.

  12. Electrical current in Sinai billiards under general small forces.
    (with Nikolai Chernov and Hong-Kun Zhang)
    J. Stat. Phys. 153 (2013), no. 6, 1065--1083. DOI; pdf.
    Main result: We prove the linear response formula for the electric currents generated by general forces (electric and nonelectric), and prove that the distribution of diffusion is normal under small external forces. Moreover, we give several characterizations of the non-equilibrium steady state of the forced system.

  13. Ergodicity of the generalized lemon billiards.
    (with Jingyu Chen, Luke Mohr and Hong-Kun Zhang)
    Chaos 23, 043137 (2013). DOI
    Main result: 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 symmetric lemon-shaped billiards. Note that there is at most one ergodic table among all lemon tables. In our asymmetric generalization, it is observed numerically that the ergodicity is no longer a rare phenomenon, but becomes prevalent. Moreover, numerical estimates of the mixing rate of the billiard dynamics on some ergodic tables are also provided.

  14. Diffeomorphisms with global dominated splittings can not be minimal.
    Proc. Amer. Math. Soc. 140 (2012), 589--593. DOI; arXiv.
    Main result: Diffeomorphisms with a dominated splitting appear in the study of robust dynamical properties and have been studied very well recently. We prove that any map with a dominated splitting cannot be minimal. Equivalently, if a map is minimal, then it cannot have any dominated splitting. Using the same aregument one can prove that if the map admits a dominated splitting, then it is not uniquely ergodic.

  15. Partially hyperbolic sets with positive measure and ACIP for partially hyperbolic systems.
    Discrete Continu. Dynam. Syst. 32 (2012), 1435--1447. DOI; arXiv.
    Main result: We study the transitivity property of partially hyperbolic systems. We prove that if such a map \(f\) is accessible and admits an ACIP, then the map \(f\) is (physically) transitive. That is, the set of points with dense orbits has positive volume. Moreover, if \(f\) is center bunched, then we have the following dichotomy: either \(f\) preserves a smooth measure, or there is no ACIP at all.

  16. Dimension of stable sets and scrambled sets in positive finite entropy systems.
    (with Chun Fang, Wen Huang and Yingfei Yi)
    Ergod. Th. Dynam. Syst. 32 (2012), no. 2, 599--628. DOI
    Main result: We study the dimensions of stable sets and scrambled sets of a dynamical system with positive finite entropy. We show that there is a measure-theoretically large set containing points whose sets of hyperbolic points (i.e. points lying in the intersections of the closures of the stable and unstable sets) admit positive Bowen dimension entropy; under the continuum hypothesis, this set also contains a scrambled set with positive Bowen dimension entropy.

  17. Pointwise dimension, entropy and Lyapunov exponents for \(C^1\) maps.
    (with Wen Huang)
    Trans. Amer. Math. Soc. 364 (2012), no. 12, 6355--6370. DOI
    Main result: For a general dynamical system, there are many invariant measures, and most of these invariant measures are not smooth, but rather singular. Various fractal dimensions have been introduced recently to characterize how singular an invariant measure is. We give an estimate of the fractal dimensions of an ergodic measure in terms of the measure-theoretic entropy and the Lyapunov exponents for general \(C^1\) selfmaps.

  18. Exponential growth rate of paths and its connection with dynamics.
    (with Zhihong Xia)
    Progress in Variational Methods, World Scientific, 2010, 212--224. DOI
    Main result: We compute the exponential growth rate of the number of paths in an associated directed graph \(G\) with length matrix \(L\) via classifying the paths by their types of primitive cycles. We prove that the exponential growth rate \(\lambda\) of the paths equals to the topological entropy of special suspension flows associated to \(L\). Our result can be interpreted in population dynamics, where two species (dominant v.s. recessive genes) have two different reproduction periods. Then \(\lambda\) gives the asymptotic exponential growth rate of total population.