2019年“微分动力系统”小型研讨会
会议时间：2019年10月29日至11月01日
会议地点：精正楼二楼学术报告厅（10月30日下午例外，在维格堂319）
2019年10月29日报到
2019年10月30日上午 精正楼二楼学术报告厅
8:50-9:00签到、开幕
9:00-9:45
报告题目：Prevalence in Ergodic optimization, (non)-Sturiman maximizing problems for doubling maps
报告人：张一威（华中科技大学）
摘要：Given a dynamical system, we say that a performance function has property P if its time averages along orbits are maximized at a periodic orbit. It is conjectured by several authors that for sufficiently hyperbolic dynamical systems, property P should be typical among sufficiently regular performance functions. In this talk， we address this problem using a probabilistic notion of typicality that is suitable to infinite dimension: the concept of prevalence as introduced by Hunt, Sauer, and Yorke. For the one-sided shift on two symbols, we prove that property P is prevalent in spaces of functions with a strong modulus of regularity. Our proof uses Haar wavelets to approximate the ergodic optimization problem by a finite-dimensional one, which can be conveniently restated as a maximum cycle mean problem on a de Bruijin graph. If time permits, I will also discuss the applications of the above combinatorial optimization methods in dealing with (non)-Sturiman maximizing (open) problems for doubling maps.
9:50-10:35
报告题目：Statistical properties for partially hyperbolic diffeomorphisms with mostly expanding and contracting centers
报告人：糜泽亚（南京信息工程大学）
摘要：For partially hyperbolic diffeomorphisms with mostly expanding and contracting centers, we study their statistical stability. We also construct the skeletons for these diffeomorphisms, which determines the number and supports of physical measures. By further analysis on skeletons, we obtain the continuity of physical measures with respect to diffeomorphisms in C^1 topology.
10:50-11:35
报告题目：Ergodic theory for expanding Thurston maps
报告人：李智强（北京大学）
摘要：Thurston maps are a class of branched covering maps on the 2-sphere that arose in W.~Thurston's characterization of postcritically finite rational maps. By imposing a natural expansion condition, M.~Bonk and D.~Meyer investigated a subclass of Thurston maps known as expanding Thurston maps, which turned out to enjoy nice topological, metric, and dynamical properties. Contrary to what the name may suggest, these maps have very weak expansion properties.
This talk will serve as an introduction to expanding Thurston maps. We will introduce these maps with some motivation from their connection to other topics of mathematics. We will then focus on the ergodic properties, especially the thermodynamic formalism for such maps.
2019年10月30日下午 维格堂319
14:00-14:45
报告题目：Existence and finiteness of physical measures for star flows
报告人：张金华（北京航空航天大学）
摘要：In this talk, we focus on star flows named by S. Liao. We will show that Palis conjecture holds for star flows: for $C^1$-generic star flows, there exist finitely many physical measures whose basins form a full Lebesgue measure subset of the whole manifold. This is a joint work with S. Crovisier, X. Wang and D. Yang.
14:50-15:35
报告题目：Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve
报告人：韩茂安（上海师范大学）
摘要：This paper deals with the number of limit cycles for planar piecewise smooth near-Hamiltonian or near-integrable systems with a switching curve. The main task is to establish a so-called first order Melnikov function which plays a crucial role in the study of the number of limit cycles bifurcated from a periodic annulus. We use the function to study Hopf bifurcation when the periodic annulus has an elementary center as its boundary. As applications, using the first order Melnikov function, we consider the number of limit cycles bifurcated from the periodic annulus of a linear center under piecewise linear polynomial perturbations with three kinds of quadratic switching curves. And we obtain 3 limit cycles for each case.
16:00-16:45
报告题目：Exact peakon, periodic peakon and pseudo-peakon solutions for the rotation-two-component Camassa-Holm system
报告人：李继彬（华侨大学）
摘要：We apply the method of dynamical systemsdevelopedby [Li \& Chen, 2007] torotation-two-component Camassa-Holm system. Through qualitative analysis, under given parameter conditions, we obtain exact explicit solitary wave solutions, pseudo-peakon solution, peakon and periodic peakon, as well as compacton solutions. To guarantee the existence of these solutions, we give constraint parameter conditions.
2019年10月31日上午 精正楼二楼学术报告厅
9:00-9:45
报告题目：Odd-odd continued fractions
报告人：廖灵敏（Université Paris-Est Créteil Val de Marne）
摘要：We define a new continued fraction algorithm called odd-odd continued fraction, whose principal convergents are rational numbers of odd denominators and odd numerators. It is obtained by an accelerated algorithm of Romik's dynamical system.
We prove that all the best approximating rationals of odd denominators and odd numerators of an irrational number is given by the principal convergents of the odd-odd continued fraction algorithm and vice versa. This is a joint work with Dong Han KIM and Seul Bee LEE.
9:50-10:35
报告题目：Skew products with concave maps
报告人：Michal Rams（波兰科学院）
摘要：We consider a special skew product of two concave maps. This system is rather interesting for a number of reasons, in particular it is open in parameter space and also its simplex of invariant measures is locally isolated - hence this system can appear as a part of a larger whole and still can be investigated in separation.
The system involves different types of hyperbolicity, including the possibility of nonuniform hyperbolicity. We investigate the spaces of invariant and of ergodic measures, especially the question of approximation of nonuniformly hyperbolic measures with uniformly hyperbolic measures. We also investigate the bifurcations by which we can exit this class of systems. From the symbolic dynamics point of view, the space of admissible trajectories form a new type of coded systems, which does not seem to appear in the literature before.
This is a joint work with Lorenzo Diaz from PUC-Rio and Katrin Gelfert from UFRJ.
10:50-11:35
报告题目：Yet another mass transference principle for arbitrary shapes.
报告人：Tomas Persson （Lund University）
摘要：The so called mass transference principle is an important tool to calculate the Hausdorff dimension of certain sets that appear in metric number theory and other settings. Consider a sequence of balls and the set of points that are contained in infinitely many of balls. Suppose that this set has full Lebesgue measure. Then themass transference principle gives an explicit lower bound of the Hausdorff dimension of this set if the balls are replaced by balls of smaller diameters but same centres.
Recently, Henna Koivusalo and Michał Rams proved a mass transference principle which estimates the Hausdorff dimension if the balls are replaced by arbitrary open subsets instead of just smaller balls.
In this talk, I will prove something similar but with a different method.
2019年10月31日下午 精正楼二楼学术报告厅
14:00-14:45
报告题目：Hyperbolicity vs. non-hyperbolic ergodic measures inside homoclinic classes
报告人：王晓东（上海交通大学）
摘要：We characterize the hyperbolicity of a homoclinic class through the hyperbolicity of ergodic measures supported on it. To be precise, we prove that for generic diffeomorphisms, a homoclinic class either is hyperbolic or supports a non-hyperbolic ergodic measure. This proves a conjecture raised by Diaz-Gorodetski. We also explore the support and the number of vanishing Lyapunov exponents of the non-hyperbolic ergodic measure under certain conditions.
14:50-15:35
报告题目：Livsic theorems for Banach cocycles
报告人：邹瑞（南京信息工程大学）
摘要：We prove a nonuniformly hyperbolic version Livsic theorem, with cocycles taking values in the group of invertible bounded linear operators on a Banach space. The result holds without the ergodicity assumption of the hyperbolic measure.Moreover, we also prove a $\mu$-continuous solution of the cohomological equation is actually Holder continuous for the uniform hyperbolic system.
16:00-16:45
报告题目：Some statistical properties of almost Anosov diffeomorphisms.
报告人：张旭（山东大学）
摘要：We talk about the existence of Sinai-Ruelle-Bowen invariant measures for a class of higher-dimensional almost Anosov diffeomorphisms. Based on the existence of invariant measures, we consider the estimates of decay rates of the correlation funcitons for Holder observables.
2019年11月1日 离会