Star systems are ones that have robustly no non-hyperbolic critical elements (singularity or periodic orbit). Since structural stability implies the star condition, a main step in studying the stability conjecture is to work on star systems. Hayashi (1992) showed that star diffeomorphisms satisfy Axiom A. However, the Lorenz attractor shows that star flows may not satisfy Axiom A if the system has a singularity. This highlights the significance of the role of singularities for flows.
In [Li-Gan-Wen, DCDS (2005)], Gan, Wen and Li introduced the extended linear Poincaré flow to study flows with singularities. They are able to deal with singularities and regular points simultaneously, for instance for star flows or flows with partial hyperbolicity. Both the final singular-hyperbolicity description of star flows and the solution of Palis weak density conjecture for 3-D flows rely heavily on this technique.
In [Gan-Wen, Invent. Math. (2006)], Gan and Wen proved that every nonsingular star flow satisfies Axiom A and the no-cycle condition. This implies the Omega stability conjecture stating that Omega stability implies Axiom A, and in turn implies the stability conjecture stating that the structural stability implies Axiom A.
They went on to characterize the singular hyperbolicity of star flows [Shi-Gan-Wen, JMD (2014)], which was proven to the best description by the recent example of Bonatti-da Luz.
In [Wang-Sun, Trans. Amer. Math. Soc. (2010)], with his student, he proved that the Lyapunov exponents of hyperbolic measures can be approximated by that of periodic measure. This is a first result on Lyapunov exponents approximation in both uniformly hyperbolic systems and non-uniformly hyperbolic systems.
In [Liang-Liao-Sun, Proc. Amer. Math. Soc. (2014)], with his students, he proved that hyperbolic measures can be approximated by periodic measures.
In [Sun-Todd-Zhou, Trans. Amer. Math. Soc. (2009)], with his co-authors, he constructed equivalent smooth flows such that one has zero entropy and the other has positive entropy. This solves an open problem by Ohno in 1980. The flow constructed in their paper is likely the “black hull” in mathematical version.
In [Sun-Zhang-Zhou, Topology Appl. (2016)], with his students, he constructed equivalent topological flows such that one has zero entropy and infinite growth rate of period and the other has infinite entropy and zero growth rate of period. This shows the extreme degeneracy for entropy and growth rate of period.
In [Liang-Liao-Sun-Tian, Trans. Amer. Math. Soc. (2017)], with his students, he established a variation principle for non-uniformly hyperbolic systems.
2.
Differential Equations (LIU Bin, LI Weigu, YANG Jiazhong)
1) Hamiltonian Systems, KAM Theory, and
Quasi-periodic Solutions for Differential Equation
In [Liu, J. Differential Equations (2009)], Bin Liu considered a
bounded perturbation of a class of forced isochronous oscillators with
repulsive singularity. Under a Lazer-Landesman type condition combined with
other regular assumptions on the associated potential function, he proved the
boundedness of all solutions as well as the existence of infinitely many
quasi-periodic solutions.
In [Capietto-Dambrosio-Liu, Z. Angew. Math. Phys. (2009)], with his
co-authors, Liu proved all the solutions are bounded for a class Duffing-type
equations with some class of singular potentials, which answered a question of
Littlewood in the early 1960's for these equations.
In [Jin-Liu-Wang, J. Math. Anal. Appl. (2011)], with his co-authors, Liu proved there
exists infinitely many quasiperiodic solutions for a class of coupled
Duffing-type equations.
In [Fu-Liu-Mi, Acta Math. Hungar. (2015)], with his co-authors, they showed the
exact asymptotic behavior of the unique solution for some singular boundary
value problem.
In [Huang-Li-Liu, Nonlinearity (2016)], with his co-authors, they proved there exists
infinitely many quasiperiodic solutions for a class of asymmetric oscillators,
and all these solutions are bounded.
2)
Rotation Numbers, Limit Cycles, Normal Forms and Linearization Theories for
Differential Equations
In [Li-Lu, Trans. Amer. Math. Soc. (2008)], Prof. Weigu Li (and K. Lu) introduced a
concept of rotation number for lifts of random orientation-preserving
homeomorphisms on the circle, not necessarily on [0, 1]. The construction also
works for continuous systems generated by random ODE's. They gave conditions under
which there is an analytical random conjugacy with a pure rotation given by
this rotation number itself.
In [Wu-Li, J. Differential Equations (2008)],
he (and H. Wu) proved an analogue of the Poincaré normal form theorem for
non-autonomous systems using a homotopy method. In [Li-Llibre-Wu, Ergodic Theory Dynam. Systems (2009)], he (and J. Llibre, H. Wu) studied
normal forms for almost periodic differential systems.
In [Li-Llibre-Yang-Zhang, J. Dynam. Differential Equations (2009)],
he (and J. Llibre, J. Yang, Z. Zhang) gave a sharp upper bound for the maximum
number of limit cycles bifurcating from the period annulus of any homogeneous
and quasi-homogeneous center.
In [Li-Lu, Discrete Contin. Dyn. Syst. Ser. B (2016)], he (and K. Lu) studied questions
related to smooth linearization of random dynamical systems when a zero
Lyapunov exponent exists and proved a version of the Takens Theorem for random
diffeomorphisms.
3) Normal Form Theory and Linearization Theory
Jiazhong
Yang is interested in normal form theory and linearization theory, esp. for
vector fields and diffeomorphisms. His research interests also focus on
qualitative analysis of planar polynomial differential systems with emphasis on
limit cycles, bifurcations theory, integrabilities and center-focus problems.
In
[Li-Liu-Yang, J. Differential Equations (2009)], Li-Liu-Yang studied the
number and configuration of limit cycles of cubic system. This is related to
the well-known Hilbert’s 16th problem. For more than more century,
the topic of limit cycles remains to be one of the most interesting subjects.
In this paper, the authors theoretically proved that there exist cubic systems
which can have up to 13 limit cycles.
The
center-focus problem is one of the most classical problems in o.d.e. To
distinguish a focus from a center-like system is extremely difficult. In [Qiu-Yang, J. Differential Equations (2009)], by an explicit construction, Qiu-Yang
gave a polynomial system having a linear center of degreen, which can reach as high asn2 of focus order. This result
negatively convinced us that integrability problem is far from trivial.
When a
polynomial system admit a center region, i.e., its orbits are closed curves in
the region, then it is very interesting to consider the critical values of the
period function. In [Gasull-Liu-Yang, J. Differential Equations (2010)], Gasull-Liu-Yang showed by
giving a concrete example that the period function of polynomial systems can be
extremely complicated since the critical values of the system can of be the
order ~n2.
In [Liu-Chen-Yang, Nonlinearity (2012)], Liu-Chen-Yang considered the existence of
hyperelliptic limit cycles of Lienard systems. The latter has very important
applications in various fields and has been widely studied. In this paper, they
obtained a complete classification of such systems.
In [Dong-Liu-Yang, Qual. Theory Dyn. Syst. (2015)], Dong-Liu-Yang considered the so-called
generalized center focus problem and gave an estimation of highest possible
saddle order. This result will have certain positive influence on further study
of integrability problem, especially in the category of complex systems.
3.
Smooth Ergodic Theories (SUN Wenxiang, LIU Peidong, SHU Lin)
Wenxiang Sun is interested in smooth
ergodic theories; in particular, he focuses on the ergodic theory of
nonuniformly hyperbolic systems. With his co-authors, he solved several open
problems proposed by Bowen, Walters, and Liao.
In [Wang-Sun, Trans. Amer. Math. Soc. (2010)], with his student, they proved that
the Lyapunov exponents of hyperbolic measures can be approximated by that of
periodic measure. This is a first result on Lyapunov exponents approximation in
both uniformly hyperbolic systems and non-uniformly hyperbolic systems.
In [Liang-Liao-Sun, Proc. Amer. Math. Soc. (2014)], with his students, they proved that
hyperbolic measures can be approximated by periodic measures.
In [Sun-Todd-Zhou, Trans. Amer. Math. Soc. (2009)], with his co-authors, they
constructed equivalent smooth flows such that one has zero entropy and the
other has positive entropy. This solves an open problem by Ohno in 1980. The
flow constructed in our paper is likely the “black hull” in mathematical
version.
In [Sun-Zhang-Zhou, Topology Appl. (2016)], with his students, they constructed
equivalent topological flows such that one has zero entropy and infinite growth
rate of period and the other has infinite entropy and zero growth rate of
period. This shows the extreme
degeneracy for entropy and growth rate of period.
In [Liang-Liao-Sun-Tian, Trans. Amer. Math. Soc. (2017)], with
his students, they established a variation principle for non-uniformly
hyperbolic systems.
Peidong Liu is interested in entropy
formulas and conjectures related to Sinai-Ruelle-Bowen
(SRB) measures, which are a class of important measures of physical
significance in dynamical systems.
In [Liu, Comm.
Math. Phys. (2008)],
he gave an equality relating entropy, folding entropy and negative Lyapunov exponents for
a non-invertible map of a finite-dimensional manifold and showed that the
equality holds if and only if the invariant measure have smooth conditional
measures on the stable manifolds. In [Liu-Shu, Nonlinearity (2011)],
he and L. Shu investigated
the entropy production of a non-invertible dynamical system and showed it is
zero if and only if the invariant measure is absolutely continuous with respect
to Lebesgue measure.
In [Liu-Lu, Discrete Contin. Dyn. Syst. (2015)], he and K. Lu showed that Shub’s entropy conjecture is true
for a partially hyperbolic attractor in a finite dimensional manifold and
proved the existence of SRB measures by using random perturbations. In
[Lian-Liu-Lu, J. Differential Equations (2016)], he
and Z. Lian, K. Lu considered partially hyperbolic attractors of a discrete-time
dynamical system in a Hilbert space. They showed the existence of SRB measures
and also investigated their ergodic properties.
Dr. Lin Shu is interested in dimension
theories and rigidity problems.
Eckamann-Ruelle
conjectured that an ergodic measure on a compact Riemannian manifold without
boundary, preserved by aC2 endomorphism, is exact dimensional.
She confirmed this and gave a new Lyapunov dimension formula for endomorphisms
[Shu, Comm. Math. Phys. (2010); Shu, Comm.
Math. Phys. (2009)].
In [Ledrappier-Shu, Trans. Amer. Math. Soc. (2014)], she (and F. Ledrappier) used the linear
drift and stochastic entropy in the universal cover space to characterize the
locally symmetric property of a manifold without focal points. In [Ledrappier-Shu, Ann. Inst. Fourier (Grenoble) (to appear)], they continued to study the differentiability of
these quantities under conformal metric changes. In particular, they gave the
formula of the differentials and showed the locally symmetric metrics are
critical points of the linear drift and entropy.
