Dimension properties of Deterministics and Stochastic fractal sets

Fall 2017

Web page: http://www.math.toronto.edu/ilia/MAT1006.2017/.

Class Location & Time: Mon, 1:00 - 2:00 PM and Wed, 12:00 PM - 2:00 PM; BA6180
Extra classes: September 25, October 16, October 30: 12:00-1:00pm; BA6180
No classes October 23 and 25

Instructor: Ilia Binder (ilia@math.toronto.edu), BA6118.

Textbooks:

1. C. Bishop and Y. Peres, Fractals in Probability and Analysis, Cambridge University Press, Dec 22, 2016
2. M. Zinsmeister, Thermodynamic Formalism and Holomorphic Dynamical Systems, American Mathematical Society; (November 1999)

Course presentation

Course notes:

1. Minkowski dimension.
2. Hausdorff dimension and measure.
3. Self-similar attractors.
4. Furstenberg Lemma.
5. Packing dimension.
6. Frostman Lemma.
7. Products and Slices.
8. Potential Theory and Dimension.
9. Projections.
10. Random Cantor sets.
11. Billingsley Lemma and its applications.
12. Dimension of a measure and Eggleston's Theorem.
13. Shift as a model dynamical system.
14. Pressure and topological entropy.
15. Perron-Ruelle-Frobenius Theorem.
16. Gibbs measures.
17. Conformal expanding repellers.
18. Pressure spectrum and Multifractal Analysis.
19. An application to Complex Dynamics.
20. Dimension of Harmonic measure for simply connected planar domains.
21. Martingales.
22. Bloch martingales and Makarov's law of iterated logarithm.
23. Löwner Evolution.
24. Slit domains.