威尼斯人

5月6日

5月7日

摘要：

题目：The rigidity of reducibility of analytic or Geverycocycles

报告人：侯宣继

摘要：We consider the rigidity problem of cocycles on somecompact groups. We prove that in generic sense an analytic or Gevery cocycle isreducible in analytic or generic sense respectively as long as it is reduciblein measurale sense and the cocycle is sufficiently close to constants inanalytic or gecery sense respectively.

题目：Approximation of Bernoulli measures for non-uniformlyhyperbolic systems

报告人： 廖刚
摘要：An invariant measure is called a Bernoulli measureif the corresponding dynamics is isomorphic to a Bernoulli shift. We prove thatfor $C^{1+/alpha}$ diffeomorphisms any weak-mixing hyperbolic measure could beapproximated by Bernoulli measures. This also holds true for $C^1$diffeomorphisms preserving a weak-mixing hyperbolic measure with dominatedOseledets decomposition.

题目：A Large Deviation Theorem for Random Diffeomorphismson Surfaces
报告人： 刘小川
摘要：In this work, we deal with random volume-preservingdiffeomorphisms. We put a mild condition on a probability measure on the groupof volume-preserving diffeomorphisms of a surface, under which we can show thelarge deviation of almost every starting point towards the volume measure. Wealso give some implications, including a version of law of large numbers, aswell as the case of perturbation of random linear transformations on 2-torus.In this talk, we will focus on the proof of the large deviation theorem, we willalso explain the tools we need, including some Benoist Quint theory, somerecent results on measure rigidity, and classical tools in smooth ergodictheory.

题目： Absolutely continuous spectrum of multi-frequencyQuasiperiodic Schrodinger operator

报告人： 王婧

摘要： We prove that for any d-frequency analyticquasiperiodic Schrodinger operator, if the  frequency is weak Liouvillean,and the potential is small enough, then the corresponding operator hasabsolutely continuous spectrum.  If d=2, we  establish the existenceof ac spectrum under small potential and some super-liouvillean frequency,  whichis optimal due to recent counterexample of Avila and Jitomirskaya.

题目：Some results of Hamiltonian homeomorphism onaspherical closed surfaces.

报告人：王俭

摘要：On closed symplectically aspherical manifolds, Schwarzproved a classical result that the action function of a nontrivial

Hamiltonian diffeomorphism is not constant by using Floer homology. Inthis article, we generalize Schwarz's theorem to the $C^0$-case

on closed aspherical surfaces. Our methods involve the theory of transversefoliations for dynamical systems of surfaces inspired by

Le Calvez and its recent progresses. As an application, we prove that thecontractible fixed points set (and consequently the fixed points

set) of a nontrivial Hamiltonian homeomorphism is not connected. We alsoget a similar result of an area preserving and orientation preservinghomeomorphism of the two sphere by applying Brouwer plane translation theoremand a recent result by F. Beguin, S. Crovisier, F. Le Roux.

题目：Hyperbolicity versus weak periodic orbits insidehomoclinic classes

报告人： 王晓东
摘要：We prove that, for C1-generic diffeomorphisms, ifthe periodic orbits contained in a homoclinic class H(p) have all theirLyapunov exponents bounded away from 0, then H(p) must be (uniformly)hyperbolic. This is in sprit of the works of the stability conjecture, but witha significant difference that the homoclinic class H(p) is not known isolatedin advance, hence the “weak” periodic orbits created by perturbations nearthe  homoclinic class have to beguaranteed strictly inside the homoclinic class. In this sense the problem isof an “intrinsic” nature, and the classical proof of the stability conjecturedoes not pass through.

题目：Response solution of NLS with Liouvillean Frequency 

报告人：徐新冬

摘要：Based on the modified infinite dimensional KAM theory, we construct the  $C^/infty$ smooth response solutions of analyticnonlinear  Schr/"odinger equation with $2$-dimension non-super Liouvilleanfrequency.