Curvature measure for fractals and application in pattern recognition

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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.