CM524: Hausdorff Measures and Fractals

 

 

1.      Generalities on measure theory. Outer measures. Regular outer measures. Metric outer measures. Caratheodory's lemma. Constructions of outer measures.

2.      Souslin sets.

3.      Hausdorff measures. Covering properties of Hausdorff measures. Hausdorff dimen-sion. Comparison with Lebesgue measure. Computation of Hausdorff measure and dimension. Net measures.

4.      Density properties of Hausdorff measures. Density bounds. Angular densities.

5.      Structure of sets of integral and of non-integral dimension. Tangency properties.. Projection properties.

6.      Complex analytic dynamics. Julia sets. The geometry of Julia sets. Mandelbrot sets.

7.      Generalities about fractals.

 

 

Bibliography

 

1.        Eggleston  H.G.,  Convexity, Cambridge University Press, 1958.

2.        Falconer  K. J., The Geometry of Fractal Sets, Cambridge University Press, 1985.

3.        Hewitt E. and Stromberg K, Real and Abstract Analysis, Springer Verlag, Berlin, Heidelberg, New York, 1969.

4.        Kingman J. F. C. and Taylor S.J., Introduction to Measure and Probability, Cambridge University Press,  1966.

5.        Rogers C. A., Hausdorff Measures, Cambridge University Press, 1970.

6.        Kessler P.  Hausdorff Measures. Lecture Notes delivered at Univ. of Craiova, 2002.

 

 

Prof. Peter Kessler