Fractals

The philosophy and scope of fractal geometry, Scaling and self similarity, Hausdorff measure and dimensions, Box-counting dimensions, Techniques for calculating dimensions, Local structure and projections of fractals, The Thermodynamic Formalism: Pressure and Gibb’s measures, the dimension formula, Invariant measuresand the transfer operator, Entropy and the Variational principle; The ergodic theorem; The renewal theorem; Martingales and the convergence theorem, Bi-Lipschitz equivalence of fractals; Multifractal Analysis; Applications of fractals: Iterated function systems (IFS) and Recurrent IFS, Applications to image compression, Julia sets and the Mandelbrot set, Random fractals, Brownian motion and Random walks, Percolation, Fractal interpolation,