Fractal Geometry

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Hausdorff dimension, box counting dimension, iterated function systems, continued fractions, number theory, Julia sets.

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Topic Outline: 
  • Hausdorff Measure and Dimension Definition and properties of Hausdorff measure and dimension, calculation of Hausdorff dimension for simple examples
  • Box Counting Definition of the box counting dimension, properties and problems of the box counting dimension
  • Techniques for Calculating Dimensions Basic methods, subsets of finite measure, potential theoretic methods
  • Fractals Defined by Transformations Iterated functions systems, dimensions of self similar sets, self-affine sets, applications to encoding images
  • Other Examples Continued fractions, diophantine approximations, graphs of functions dimension of graphs
  • Dynamical Systems Repellers and iterated functions schemes, the logistic map, continuous dynamical systems
  • Iteration of Complex Functions-Julia Sets General theory of Julia sets, the Julia sets of quadratic functions, the Mandelbrot set
  • Random Fractals Random Cantor sets, fractal percolation