Dynamics seminar
Student No.:40
Time:Fri 15:10-16:10, Oct.26-Jan.18
Instructor:Xue Jinxin, Huang Guan, Wang Lin  
Place:Lecture Hall, Jin Chun Yuan West Bldg.
Starting Date:2018-10-26
Ending Date:2019-1-18


Speaker: Zhiqiang Li, Stony Brook

Title: Dynamical zeta functions and prime orbit theorems in complex dynamics

Analogues of the Riemann zeta function were first introduced into geometry by A.~Selberg and into dynamics by M.~Artin, B.~Mazur, and S.~Smale. Analytic studies of such dynamical zeta functions yield quantitative information on the distribution of closed geodesics and periodic orbits.
We obtain the first Prime Orbit Theorem, as an analogue of the Prime Number Theorem, in complex dynamics outside of hyperbolic maps, for a class of branched covering maps on the $2$-sphere called expanding Thurston maps $f$. More precisely, we show that the number of primitive periodic orbits of $f$, ordered by a weight on each point induced by a non-constant real-valued H\"{o}lder continuous function on $S^2$ satisfying some additional regularity conditions, is asymptotically the same as the well-known logarithmic integral, with an exponentially small error term. Such a result follows from our quantitative study of the holomorphic extension properties of the associated dynamical zeta functions and dynamical Dirichlet series.
In particular, the above result applies to postcritically-finite rational maps whose Julia set is the whole Riemann sphere. Moreover, we prove that the regularity conditions needed here are generic; and for a Latt\`{e}s map $f$, a continuously differentiable function satisfies such a condition if and only if it is not cohomologous to a constant. This is a joint work with T.~Zheng.


Speaker: Qiu Yu 邱宇,清华大学

Title: Moduli spaces of quadratic differentials on decorated marked surfaces

First we follow Bridgeland-Smith to introduce certain moduli spaces of quadratic differentials on marked surfaces, which are used to realize spaces of stability conditions. Then we discuss various framed versions and show that a proper framed version is simply connected. (This is a joint work with A. King).


Speaker: Wei Qiaoling 魏巧玲 (首都师范大学)

Title: On deformational spectral rigidity of convex planar domains

This talk is related to the famous question "Can you hear the shape of drum?" by M.Kac in 1960'.
That is, whether a planar domain Ω can be uniquely determined by its Laplace spectrum consisting of eigenvalues of a Dirichlet problem.
In general, there are counterexamples.
Meanwhile, from dynamical aspect, there is length spectrum consisting of perimeters of all periodic orbits of a billiard problem inside Ω.
The Laplace and length spectra are closely related, generically the first determines the second.
During the talk we show that a planar axis symmetric domain close to the circle can not be smoothly deformed preserving the length spectrum unless the deformation is a rigid motion.
This is a joint work with J. De Simoi and V.Kaloshin.


Speaker: Yitwah Cheung, Tsinghua University

Title: Nonuniform Excursions in Dynamics on Moduli Spaces

This talk has two parts. In the first part, I will give a brief survey of some key developments in the study of Teichmuller flows leading up to Eskin-Mirzakhani's classification of orbits of the SL(2,R)-action on the Hodge bundle. In the second part, I will explain an elementary construction of minimal non uniquely ergodic systems dating back to Veech that played a role in the inception of the field and appears to be critical to our understanding of horocycle orbits. Along the way, I will try to mention some interesting open questions and efforts to extend the general theory to include broader classes of moduli spaces.


Speaker: Zhou Qi 周麒 (南京大学)

Title: Exponential Dynamical Localization: Criterion and Applications

We give a criterion for exponential dynamical localization in expectation (EDL) for ergodic families of operators acting on $\ell^2(\Z^d)$.
As applications, we prove EDL for a class of quasi-periodic long-range operators on $\ell^2(\Z^d)$.
We also prove EDL for a class of quasi-periodic Schr\"odinger operator on $\ell^2(\Z^d)$ with almost sharp dynamical localization exponent.
For supercritical almost Mathieu operators, we prove EDL with sharp dynamical localization exponent. This is joint work with J.You and L.Ge.


Speaker: 张建路Zhang Jianlu(中科院数学所)

Title: Asymptotic density of the collision orbits in the restricted planar circular 3 body problem

For the Restricted Circular Planar 3-Body Problem, we show that there exists a full dimensional open set U in phase space independent of the mass ratio μ, where the set of initial points which lead to collision is O(μ ^1/20 ) dense as μ → 0. It partially supports the conjecture of Alexseev in 1970s.


Speaker: Hiroyoshi Mitake (The University of Tokyo)

Title: On large-time behavior for birth-spread type nonlinear PDEs

In this talk, we introduce birth-spread type nonlinear partial differential equations which is motivated by a crystal growth phenomenon. Mathematically, an interesting nonlinear phenomenon in terms of asymptotic speed of solutions appears which is sensitive to the shapes of source terms. We discuss properties of large-time asymptotic speed, and also present recent results of large-time asymptotic profile of solutions.
This is a joint work with Y. Giga (Univ. of Tokyo) and H. V. Tran (Univ. of Wisconsin-Madison).


Speaker: Alxandre Boritchev [Université Claude-Bernard (Lyon I)]

Title: Exponential convergence and hyperbolicity of the minimisers for random Lagrangian systems

We consider the stochastic Burgers equation from the Lagrangian point of view (long-time behaviour of the minimisers) as well as from the point of view of the statistical behaviour of the solutions (long-time convergence towards the stationary measure). In both cases there is a phenomenon of exponential convergence.
Part of the presentation is about a joint work with K. Khanin (University of Toronto)font>