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Weekly seminar links: Dynamical Systems and ODEs

##### Dynamical Systems and ODEs

**1. Differential Dynamical Systems (WEN Lan, SUN Wenxiang, GAN Shaobo, SHI Yi)**

**1) Star Flows **

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.

**2) Palis Weak Density Conjecture **

A long-standing goal in dynamical systems since S. Smale is to characterize the behavior of most systems. To this end, Palis raised a series of density conjectures, one of which states that every system can be approximated either by Morse-Smale systems or by systems exhibiting Smale’s horseshoe. Gan and Wen proved this density conjecture for 3-D diffeomorphisms in [Bonatti-Gan-Wen, ETDS (2007)]. The method played a key role in the final solution of the conjecture for diffeomorphisms [Crovisier, Ann. of Math. (2) (2010)]. By using the technique of the extended linear Poincaré flow in [Li-Gan-Wen, DCDS (2005)], they proved this density conjecture for 3-D flows in [Gan-Yang, ASENS (to appear)].

**3) Entropy and Nonuniformly Hyperbolic Systems **

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, 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.

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