Deterministic and random iterated function systems

确定性和随机迭代函数系统

• 批准号: 327304-2006
• 负责人: Roy, Mario
• 金额: $ 0.66万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2006
• 资助国家: 加拿大
• 起止时间: 2006-01-01 至 2007-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=339969
• 关键词: Deterministic; random; iterated; function; systems

Iterated function systems (abbreviated to IFSs) arise in many natural contexts. They are often used to encode and generate fractal images, such as landscapes and skyscapes, in computer games. They each generate, via a recursive procedure, a unique fractal set. They also play an important role in the theory of dynamical systems, and are in fact a generalisation of the process of looking at the backward trajectories of physical systems. My research aims at studying deterministic and random IFSs. Random IFSs are similar to their deterministic counterparts, but their generators are chosen randomly. The non-uniform outlook of the fractal sets they generate is usually more reminiscent of natural objects such as trees, clouds or mountains. In the 1980's and 1990's, pioneering work was accomplished on IFSs which comprise finitely many generators, each of these generators being a similarity map, that is, a transformation that preserves shapes. My research work concentrates on IFSs which are generated by finitely or even infinitely many transformations that do not necessarily preserve shapes, but that at least preserve angles. This type of IFSs has already found and will hopefully lead to new applications in number theory, complex dynamics and non-commutative algebra.

迭代函数系统（简称IFS）出现在许多自然环境中。它们通常用于编码和生成分形图像，例如计算机游戏中的风景和天空。他们每个人都产生，通过递归过程，一个独特的分形集。它们在动力系统理论中也起着重要的作用，实际上是对物理系统向后轨迹研究过程的概括。我的研究旨在研究确定性和随机IFS。随机IFS类似于它们的确定性对应物，但它们的生成器是随机选择的。它们生成的分形集的不均匀外观通常更容易让人联想到自然物体，如树木，云或山脉。在20世纪80年代和90年代，在IFS上完成了开创性的工作，IFS包括许多生成器，这些生成器中的每一个都是相似性映射，即保持形状的变换。我的研究工作集中在IFS的产生，不一定保持形状，但至少保持角度，甚至无穷多的转换。这类迭代函数系统已经在数论、复动力学和非交换代数等领域得到了广泛的应用。