Publications of Pengfei Zhang
Papers

Homoclinic points of symplectic partially hyperbolic systems with 2D centre.
Dynamical Systems 39 (2), 302316, 2024; DOI
arXiv.
Abstract: We study some generic properties of symplectic partially
hyperbolic systems with 2D center. We prove that every hyperbolic periodic point has transverse homoclinic intersections for a generic symplectic partially hyperbolic diffeomorphism close to direct/skew products of symplectic Anosov diffeomorphisms with areapreserving diffeomorphisms.

Homoclinic and heteroclinic intersections for lemon billiards.
(joint with Xin Jin)
Advances in Mathematics 442, April 2024, 109588; (DOI);
arXiv.
Abstract: We study the dynamical billiards on a symmetric lemon table \(\mathcal{Q}(b)\), where \(\mathcal{Q}(b)\) is the intersection of two unit disks with center distance \(b\). We show that there exists \(\delta_0 > 0\) such that for all \(b\in (1.5, 1.5+\delta_0)\) (except possibly a discrete subset), the billiard map \(F_b\) on the lemon table \(\mathcal{Q}(b)\) admits crossing homoclinic and heteroclinic intersections. In particular, such lemon billiards have positive topological entropy.

Birkhoff Normal Form and Twist Coefficients of Periodic Orbits of Billiards.
(joint with Xin Jin)
Nonlinearity 35 (8) (2022) 39073943; DOI; arXiv.
Abstract: In this paper we study the Birkhoff Normal Form around elliptic periodic points for a variety of dynamical billiards. We give an explicit construction of the Birkhoff transformation and obtain explicit formulas for the first two twist coefficients in terms of the geometric parameters of the billiard table. As an application, we obtain characterizations of the nonlinear stability and local analytic integrability of the billiards around the elliptic periodic points.
Codes
Notebook Archive

Decay of correlations for unbounded observables.
(joint with Fang Wang and HongKun Zhang)
Nonlinearity 34 (4) (2021), 24022429.
DOI;
arXiv
Abstract: 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 semidispersing billiards on a rectangle, billiards with cusps, and Bunimovich stadia.

Hyperbolicity of asymmetric lemon billiards.
(joint with Xin Jin)
Nonlinearity 34 (1) (2021), 92117.
DOI;
arXiv
Abstract: 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 \(R\).

Homoclinic intersections for geodesic flows on convex spheres.
(joint with Zhihong Xia)
Contemporary Math. 698 (2017), 221238.
DOI;
arXiv
Abstract: 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.

Convex billiards on convex spheres.
Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), 793816.
DOI;
arXiv.
Abstract: 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 (KupkaSmale). 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.

Accessibility and homology bounded strong unstable foliation for Anosov diffeomorphisms on 3torus.
(joint with Yan Ren and Shaobo Gan)
Acta Math. Sinica 33 (2017), 7176.
DOI;
arXiv.
Abstract: 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.

On another edge of defocusing: hyperbolicity of asymmetric lemon billiards.
(joint with Leonid Bunimovich and HongKun Zhang)
Comm. Math. Phys. 341 (2016), 781803.
DOI;
arXiv.
Abstract: 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.

Homoclinic points for convex billiards.
(joint with Zhihong Xia)
Nonlinearity 27 (6) (2014), 11811192.
DOI;
arXiv.
Abstract: The only way to perturb a billiard system is to deform the billiard table. Such deformations always produce nonlocal 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.

Fundamental domain of invariant sets and applications.
Ergod. Th. Dynam. Syst. 34 (2014), no. 1, 341352.
DOI;
arXiv.
Abstract: 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.

Electrical current in Sinai billiards under general small forces.
(joint with Nikolai Chernov and HongKun Zhang)
J. Stat. Phys. 153 (2013), no. 6, 10651083.
DOI;
pdf.
Abstract: 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 nonequilibrium steady state of the forced system.

Ergodicity of the generalized lemon billiards.
(joint with Jingyu Chen, Luke Mohr and HongKun Zhang)
Chaos 23, 043137 (2013).
DOI
Abstract: 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 symmetric lemonshaped 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.

Diffeomorphisms with global dominated splittings can not be minimal.
Proc. Amer. Math. Soc. 140 (2012), 589593.
DOI;
arXiv.
Abstract: 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.

Partially hyperbolic sets with positive measure and ACIP for partially hyperbolic systems.
Discrete Continu. Dynam. Syst. 32 (2012), 14351447.
DOI;
arXiv.
Abstract: 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.

Dimension of stable sets and scrambled sets in positive finite entropy systems.
(joint with Chun Fang, Wen Huang and Yingfei Yi)
Ergod. Th. Dynam. Syst. 32 (2012), no. 2, 599628.
DOI
Abstract: We study the dimensions of stable sets and scrambled sets
of a dynamical system with positive finite entropy. We show that there is a measuretheoretically 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.

Pointwise dimension, entropy and Lyapunov exponents for \(C^1\) maps.
(joint with Wen Huang)
Trans. Amer. Math. Soc. 364 (2012), no. 12, 63556370.
DOI
Abstract: 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 measuretheoretic entropy and the Lyapunov exponents for general \(C^1\) selfmaps.

Exponential growth rate of paths and its connection with dynamics.
(joint with Zhihong Xia)
Progress in Variational Methods, World Scientific, 2010, 212224.
DOI
Abstract: 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.
Preprints on arXiv

Twist interval for twist maps.
arXiv.
Abstract: 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 nonwandering, then the twist interval of \(f\) is nondegenerate. 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).