Curvature measure for fractals and application in pattern recognition

在我们项目的前一阶段，我们能够确定相当一般的自相似和自共形集合的分形曲率。最近与动力系统遍历理论的结合，导致了更简单的证明方法，计划进一步发展。现在的主要目标是扩展到新的类，特别是v变量和图定向随机分形。此外，与分形分析及其曲率特性的关系将被研究，就像从光谱行为和热核的经典案例中所知道的那样。这里通过狄利克雷形式与势理论产生联系。我们还打算支持进一步开发一个具有应用的分形曲率近似的计算机程序。

In the previous period of our project we were able to determine fractal curvatures for rather general self-similar and self-conformal sets. The recent combination with ergodic theory of dynamical systems, which leads to simpler methods of proof, is planned to be further developed. A main aim consists now in extension to new classes, in particular, V-variable and graph directed random fractals.Furthermore, relationships to analysis on fractals and their curvature properties will be investigated, like known from the classical case of spectral behavior and heat kernels. Here connections to potential theory arise via Dirichlet forms. We also intend to support the further development of a computer program for approximations of fractal curvatures with applications.