### 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:**

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

**Course presentation**

**Course notes:**

- Minkowski dimension.
- Hausdorff dimension and measure.
- Self-similar attractors.
- Furstenberg Lemma.
- Packing dimension.
- Frostman Lemma.
- Products and Slices.
- Potential Theory and Dimension.
- Projections.
- Random Cantor sets.
- Billingsley Lemma and its applications.
- Dimension of a measure and Eggleston's Theorem.
- Shift as a model dynamical system.
- Pressure and topological entropy.
- Perron-Ruelle-Frobenius Theorem.
- Gibbs measures.
- Conformal expanding repellers.
- Pressure spectrum and Multifractal Analysis.
- An application to Complex Dynamics.
- Dimension of Harmonic measure for simply connected planar domains.
- Martingales.
- Bloch martingales and Makarov's law of iterated logarithm.
- Löwner Evolution.
- Slit domains.