Instructor: Matilde Marcolli

- Notions of Dimension
- Self-Similarity
- Measure Theory
- Sharkovsky's Ordering and Chaos
- Measures and Dimension
- Entropy and Dimension
- Menger universal spaces

- Monday January 6: the notion of dimension, dimensions and boundaries, homology, linear and smooth spaces, topological dimension, self-similarity dimension, construction of the Cantor set, dynamics on the Cantor set
- Tuesday January 7: topology and geometry of the Cantor set, metric, cylinder sets, self-similarity of the Cantor set, heuristic discussion of the Hausdorff dimension, dimensional regularization, differential forms and cohomology, densities in non-integer dimension, Weyl's law for the Laplacian on manifolds
- Monday January 13: Spin manifolds and Dirac operators, analytic properties of Dirac operators on compact manifolds and their axiomatic formulation, spectral triples, Dirac operator on the Cantor set, zeta function and dimension spectrum
- Tuesday January 14: Self-similarity as a fixed point theorem, fixed point theorem for contractions on complete metric spaces, space of compact sets with the Hausdorff metric, contractions and fixed points as self-similarity, self-similarity dimension, examples: non-uniform Cantor sets, Sierpinski gasket, Koch snowflake, Peano curve, Levy dragon curve, Minkowski curve, measure spaces, continuity of measures, outer measures and measures, Caratheodory construction
- Monday January 20: existence of non-measurable sets, Banach-Tarski paradox, Hausdorff measure and Hausdorff dimension
- Tuesday January 21: properties of the Hausdorff measure and Hausdorff dimension, Hausdorff dimension of the Cantor set, Moran condition, Vitali covering lemma, Bernoulli measures and Markov measures on shift spaces, one-dimensional Markov maps and conding by subshifts of finite type, period three implies chaos, Sharkovsky ordering, proof of Sharkovsky's theorem
- Monday January 27:
- Tuesday January 28:
- Monday February 3:
- Tuesday February 4:
- Monday February 10:
- Tuesday February 11:
- Monday February 17: reading week no class
- Tuesday February 18: reading week no class
- Monday February 24:
- Tuesday February 25:
- Monday March 2:
- Tuesday March 3:
- Monday March 9:
- Tuesday March 10:
- Monday March 16:
- Tuesday March 17:
- Monday March 23:
- Tuesday March 24:
- Monday March 30: student presentations
- Tuesday March 31: student presentations

- Kenneth Falconer, "Fractal geometry" (2nd), Wiley, 2003.
- Yakov Pesin and Vaughn Climenhaga, "Lectures on fractal geometry and dynamical systems", American Mathematical Society, 2009.
- Geoffrey R. Goodson, "Chaotic Dynamics", Cambridge University Press, 2017.
- Shlomo Sternberg, "Dynamical Systems", Dover, 2010.
- Michel L. Lapidus, Machiel van Frankenhuijsen, "Fractal Geometry, Complex Dimensions and Zeta Functions", Springer 2007
- Paul Addison "Fractals and Chaos: an illustrated course", Institute of Physics Publishing, 1997.
- Christian Beck, Friedrich Schoegl, "Thermodynamic of chaotic systems", Cambridge University Press, 1993.

- pdf The notion of dimension in geometry and algebra (Yuri I. Manin)
- pdf Period three implies chaos (by T.Y.Li and A.Yorke)
- pdf The Sharkovsky theorem: a natural direct proof (Keith Burns and Boris Hasselblatt)
- pdf Approximation of continuous Newton's method: an extension of Cayley's problem (by J.Jacobssen, O.Lewis, B.Tennis)
- pdf Chaos, fractals and statistics (S.Chatterjee and M.R.Yilmaz)
- pdf Arithmetic Quantum Chaos (Jens Marklof)
- pdf Arithmetic Quantum Chaos (Peter Sarnak)
- pdf What is quantum chaos? (Zeev Rudnick)
- pdf Multifractal analysis of Lyapunov exponent for Continued fraction and Manneville-Pomeau transformations and applications to Diophantine approximation (M.Pollicott and H.Weiss)
- pdf Multifractal of the Apollonian tiling (Dominique Simpelaere)
- pdf The multifractal analysis of Birkhoff averages and large deviations (Yakov Pesin and Howard Weiss)
- pdf A generalized multifractal spectrum of the general Sierpinski carpets (by Yongxin Gui and Wenxia Li)
- pdf Spectral triples and the geometry of fractals (by Erik Christensen, Christina Ivan, Elmar Schrohe)
- pdf Dirac operators and spectral triples for some fractal sets built on curves (by Erik Christensen, Cristina Ivan and Michael Lapidus)
- pdf Complex dimension of self-similar fractal strings and Diophantine approximations (by Michel Lapidus and Machiel van Frankenhuysen)
- pdf Fifty years of entropy in dynamics (by A.Katok)