Rigidity and Small Divisors in Holomorphic Dynamics

最简单的非线性系统是由二次多项式驱动的。也就是说，状态的“时间n”由该状态的“时间n-1”的二次多项式确定。然而，尽管经过了一个多世纪的密集研究，即使是二次公式的动力学特征仍然远未被很好地理解。例如，可以用来模拟共振现象的具有“小因子”的复二次多项式在许多情况下仍然表现出神秘的行为。在过去的三十年里，人们对二次多项式的动力学进行了广泛的研究。通常，需要来自不同数学学科的复杂工具来描述这些地图的精细动态特征。通常，会引入一套这样的工具来研究一类二次映射的动力学，但这导致了对这类非线性系统的成功研究。因此，研究二次多项式的一套有效工具为在更广泛的非线性系统领域中的广泛研究提供了基础。在这个项目中，我开发了一套来自不同数学学科的新工具，以提供对某些类型的二次多项式的动力学的全面描述。它从分析、几何和更复杂的数学机器，如重正规化和TeichMuller理论中发展了有效的技术。我将实现以下主要目标：(1)小因子：本研究的一个主要目标是介绍一种系统的方法，以获得对具有小因子的二次多项式的动力学的全面了解。这提供了这类系统在共振中心具有不稳定行为的第一个例子，其动力学行为被完全理解。二次多项式的Julia集是其动力学的不稳定轨迹。最近X.Buff和A.Cheritat的一个引人注目的结果指出，存在具有可观测(正面积)Julia集的具有小因子的二次多项式。在存在小因子的情况下，一个中心问题是确定导致可见Julia集的旋转数的算术条件。所提出的研究在这一问题上取得了重大进展。(2)双曲性的刚性和密度：表现出某种众所周知的动力学行为的二次多项式称为双曲型。有一个显著的性质，由P.Fatou在1920年的S预测，指出任何二次多项式都可以扰动到具有双曲线行为的附近的一个多项式(通过适当归一化的系数的小变化)。该项目研究了重正规化技术的一些深层分析性质，以证实对某些类型的二次多项式(康托参数集)的这一猜想。本节目建议对这一猜想进行精炼的量化(本着连分式的精神)。(3)广义Feigbaum映射：倍周期分叉是实系数二次多项式族中的一种显著现象。当人们考虑复系数的二次多项式时，会出现广泛的类似但更复杂的现象。这反映了曼德尔布罗特集的复杂结构。在1980年代S和90年代S的密集研究中，这种实系数映射的动力学特征已经得到了深入的研究，而复系数映射的动力学特征却鲜有人研究。我将与全纯动力学的顶尖专家A.Avila(里约热内卢，巴西和法国巴黎)、X.Buff(法国图卢兹)、A.Cheritat(法国波尔多)和M.Shishikura(日本京都)合作开展这一重大项目的一些部分。

The simplest non-linear systems are driven by quadratic polynomials. That is, "time n" of a state is determined by a quadratic polynomial of "time n-1" of that state. However, despite over a century of intense study, the dynamical features of even quadratic formulae remain far from well understood. For example, complex quadratic polynomials with "small divisors", which may be used to model resonance phenomena, still exhibit mysterious behaviour in many cases.There has been extensive research on the dynamics of quadratic polynomials over the last three decades. Often, sophisticated tools from different disciplines of mathematics are needed to describe the fine dynamical features of these maps. Usually, a set of such tools is introduced to study the dynamics of a type of quadratic maps, but leads to the successful study of non-linear systems of that type. Thus, an effective set of tools for the study of quadratic polynomials provide the basis of extensive research in the wider area of non-linear systems. In this project, I develop a new set of tools from different disciplines of mathematics to provide a comprehensive description of the dynamics of certain types of quadratic polynomials. This develops effective techniques from analysis, geometry, and more sophisticated mathematical machinery such as renormalisation and Teichmuller theory.I will achieve the following major goals.(1) Small divisors: A main goal of this research is to introduce a systematic approach to obtain a comprehensive understanding of the dynamics of quadratic polynomials with small divisors. This provides the first examples of such systems with unstable behavior at the center of resonance, whose dynamical behaviour is completely understood.The Julia set of a quadratic polynomial is the unstable locus of its dynamics. A recent remarkable result of X. Buff and A. Cheritat states that there are quadratic polynomials with small divisors which have observable (positive area) Julia sets. A central problem in the presence of small divisors is to determine arithmetic conditions on the rotation number that leads to observable Julia sets. The proposed research makes major advances on this problem.(2) Rigidity and density of Hyperbolicity: The quadratic polynomials that exhibit a certain well understood dynamical behaviour are called hyperbolic. There is a remarkable property, anticipated by P. Fatou in 1920's, stating that any quadratic polynomial may be perturbed to a nearby one with hyperbolic behaviour (by small changes in coefficients in an appropriate normalisation). The project studies some deep analytic properties of a renormalisation technique to confirm this conjecture for certain types of quadratic polynomials (a Cantor set of parameters). This programme suggests a refined quantitative (in spirit of continued fractions) version of this conjecture to hold. (3) Generalized Feigenbaum maps:Period doubling bifurcation is a remarkable phenomenon that appears in the family of quadratic polynomials with real coefficients. There is a wide range of analogous, but more complicated, phenomena that occur when one considers quadratic polynomials with complex coefficients. This reflects the complicated structure of the Mandelbrot set. The dynamical features of such maps with real coefficients have been deeply studied in a period of intense research in 1980's and 90's, while the ones with complex coefficients are largely unexplored. The research proposal uses renormalisation techniques and develops innovative analytical methods to present a detailed description of the dynamics of such a map near degenerate bifurcations.I will carry out some parts of this major project in collaboration with the leading experts of holomorphic dynamics: A. Avila (Rio, Brazil and Paris, France), X. Buff (Toulouse, France), A. Cheritat (Bordeaux, France), and M. Shishikura (Kyoto, Japan).