Deterministic and random iterated function systems

迭代函数系统(简称IFSS)存在于许多自然环境中。在电脑游戏中，它们经常被用来编码和生成分形图，如风景和天空。它们每一个都通过递归过程生成唯一的分形集。它们在动力系统理论中也扮演着重要的角色，实际上是观察物理系统反向轨迹的过程的推广。我的研究目的是研究确定性和随机性IFSS。随机IFS类似于它们的确定性对应项，但它们的生成器是随机选择的。它们生成的分形集的非均匀外观通常更让人联想到树木、云或山脉等自然对象。20世纪80年代的S和90年代的S在综合决策支持系统上取得了开创性的工作。综合决策支持系统由有限多个生成元组成，每个生成元是一个相似映射，即保持形状的变换。我的研究工作集中在IFS，它是由有限甚至无限多个变换生成的，这些变换不一定保持形状，但至少保持角度。这类IFSS已经被发现，并有望在数论、复动力学和非对易代数中得到新的应用。

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.