Mathematical Sciences: Weak Expansion in Real and Complex Dynamics

Graczyk项目的主要目标是研究满足弱展开性质的一维实和复动力系统的几何和测量理论性质，这些性质既可以是解析的(Collet-Eackmann条件)，也可以是拓扑性的(盒子构造)。重点是描述和解释非双曲动力学中出现的分形的几何结构。在该项目的全纯部分，P.I.特别感兴趣的是有理函数Julia集上Fatou分量的Holder正则性的动态刻画和双曲子集的持久性。Holder正则性似乎与科莱-埃克曼条件密切相关，该条件只需要沿着临界轨道指数展开。这一研究方向是由Carleson，Jones和Yoccoz在他们对属于John域的Fatou分量的动态分类所做的工作中首创的。琼斯和马卡洛夫最近的工作表明，Holder Julia集是度量小的。一般而言，对于不满足Misiurewicz-瑟斯顿条件的Julia集的Hausdorff维数，我们知之甚少。P.I.有一种利用诱导双曲性来估计二次多项式的Hausdorff维数的方法。一般说来，可能的方法包括：庞加莱级数、保角测度和Julia集的解析正则性。该项目的第二部分涉及真实的一维系统。区间上的S-单峰Collet-Eockmann映射的目的是证明它们诱导双曲性，并且它们的拓扑类和拟对称类重合。另一系列有可能取得进展的问题涉及不可逆圆映射的轨道分布的几何性质和参数空间中频率轨迹的分形结构。自然界中的许多物体和现象都可以用“分形”几何来描述，它涉及到连续尺度和分数维集的自相似性。分形形状与线和圆的规则几何形状不兼容。这里的一个很好的例子是卫星图片上看到的非常混乱的海岸线形状。这类高度复杂的集合出现在非线性系统中的吸引子、混沌或分叉行为的域中，并且通常对于理解潜在的动力学是至关重要的。本项目的目的之一是表明在许多非线性系统中-这些物理、化学和生物学中的模型现象(约瑟夫森结、电荷密度波、生态系统中的种群增长、二极管-谐振器等)-混沌区域是一个小维的分形。例如，微波场中的阻性分流约瑟夫森结或射频电场中的电荷密度波可以用周期作用力的阻尼力驱动摆的微分方程来描述。该方程的二维返回映射在包含向混沌的跃迁的参数区域内折叠为一维映射。因此，这些系统中的频率锁定、噪声和迟滞可以用临界圆映射的动力学性质来描述，本课题研究了临界圆映射的动力学性质。

Abstract Graczyk The main goal of the project is to study geometric and measure theoretic properties of one dimensional real and complex dynamical systems satisfying weak expansion properties either of analytical (Collet-Eckmann condition) or topological nature (box construction). The focus is on describing and explaining the geometric structure of fractals which arise in non-hyperbolic dynamics. In the holomorphic part of the project, the P.I. is particularly interested in the dynamical characterization of Holder regularity of the Fatou components and the persistence of hyperbolic subsets in Julia sets of rational functions. Holder regularity seems to be closely related to the Collet-Eckmann condition which requires exponential expansion only along the critical orbits. This direction of study was originated by Carleson, Jones and Yoccoz in their work on the dynamical classification of Fatou components which are John domains. The recent work of Jones and Makarov would imply that the Holder Julia sets are metrically small. Generally, very little is known about the Hausdorff dimension of Julia sets which do not satisfy the Misiurewicz-Thurston condition. The P.I. has a method of estimating Hausdorff dimension for quadratic polynomials which uses induced hyperbolicity. In general, possible methods involve: Poincare series, conformal measures and analytical regularity of Julia sets. The second part of the project concerns real 1-dimensional systems. The objective for S-unimodal Collet-Eckmann mappings of the interval is to prove that they induce hyperbolicity and their topological and quasisymmetrical classes coincide. The other series of problems in which progress is possible concerns the geometric properties of the distribution of the orbits and the fractal structure of the frequency locus in the parameter space for non-invertible circle maps. Many objects and phenomena in nature can be described through ``fractal'' geometry which involves self-similarity of consecutive scales and sets of fractional dimension. Fractal shapes are not compatible with regular geometry of lines and circles. A good example here is a very chaotic coastline shape seen on satellite pictures. Such highly complicated sets occur as the domains of attractors, locci of chaotic or bifurcational behavior in non-linear systems and are often crucial in understanding the underlying dynamics.One of the aims of this project is to show that in many non-linear systems - these model phenomena in physics, chemistry, and biology ( Josephson junction, charge density waves, population growth in ecosystems, diode-resonators, etc ) - the chaotic region is a fractal of small dimension. For example, the resistively shunted Josephson junction in microwave fields or charge-density waves in radio-frequency electric fields can be described by the differential equation of the damped driven pendulum with a periodic force. The two-dimensional return map for this equation collapses to a one-dimensional map in a parameter regime including transition to chaos. Frequency locking, noise, and histeresis in these systems can thus be described by the dynamical properties of critical circle maps, which are studied in this project.