Mathematical Sciences: Weak Expansion in Real and Complex Dynamics

数学科学：实复杂动力学中的弱展开

• 批准号: 9626874
• 负责人: Jacek Graczyk
• 金额: $ 4万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 1998-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626874&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Weak; Expansion; Real

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.

摘要Graczyk 该项目的主要目标是研究一维真实的和复杂动力系统的几何和测量理论性质，满足分析（Collet-Eckmann条件）或拓扑性质（盒结构）的弱膨胀性质。 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。在项目的全纯部分，PI。对有理函数Julia集上Fatou分量的保持器正则性和双曲子集的持久性的动力学刻画特别感兴趣。保持器正则性似乎与Collet-Eckmann条件密切相关，Collet-Eckmann条件只需要沿沿着临界轨道的指数展开。这个方向的研究起源于Carleson，Jones和Yoccoz在他们的工作中对Fatou分量的动力学分类，这些分量是John域。Jones和Makarov最近的工作暗示了保持器Julia集是度量小的。 通常，很少有人知道不满足Misiurewicz-Thurston条件的Julia集的Hausdorff维数。私家侦探提出了一种利用诱导双曲性估计二次多项式Hausdorff维数的方法。一般来说，可能的方法包括：庞加莱级数，共形测度和Julia集的分析正则性。该项目的第二部分涉及真实的1维系统。 对区间上的S-单峰Collet-Eckmann映射，证明了它们诱导双曲性，并且它们的拓扑类和拟对称类是一致的。其他一系列的问题，其中的进展是可能的关注的几何性质的分布的轨道和分形结构的频率轨迹在参数空间的不可逆的圆地图。 自然界中的许多物体和现象都可以通过“分形”几何来描述，它涉及连续尺度和分数维集合的自相似性。分形形状与直线和圆的规则几何形状不兼容。 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。这种高度复杂的集合在非线性系统中作为吸引子、混沌或分叉行为的轨迹的域出现，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，生态系统中的人口增长，二极管谐振器等）-混沌区域是一个小维的分形。例如，微波场中的反向分流约瑟夫森结或射频电场中的电荷密度波可以用受周期力驱动的阻尼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过渡的参数区域中坍缩为一维映射。 因此，在这些系统中的频率锁定，噪声和histeresis可以描述的临界圆映射，这是在这个项目中研究的动力学性质。