MAT1845HS: Introduction to Fractal Geometry and Chaos Theory

Winter 2020: University of Toronto, BA6180, Monday 5-6pm and Tuesday 10am-noon

Brief Course Description

The course will give an introduction to fractal geometry and chaotic dynamics, with an emphasis on geometric aspects. Topics covered will include various notions of fractal dimensions, fractal measures, symbolic dynamics, notions of entropy, chaos in dynamical systems and strange attractors, geometric operators (Laplacians, Dirac, etc.) on fractal objects. The course will be suitable also for students from other departments interested in applications of fractals and chaos.

ANNOUNCEMENT (March 13): Due to new directives from the university, we have to suspend in-person classes. Reading material and additional slides will continue to be posted on this class webpage. Final student presentations will have to be replaced by presentations in the form of a written summary of your chosen reading material (or possibly a remote oral presentation if some convenient method is available)

Slides of Lectures

Notions of Dimension
Self-Similarity
Measure Theory
Sharkovsky's Ordering and Chaos
Measures and Dimension
Entropy and Dimension
Entropy and Dynamics
Menger universal spaces
Continued fractions and modular symbols
Schottky Groups and Limit Sets
Graphs: Random, Chaos, Quantum (joint talk for the MAT1845HS class and the Fields Institute Focus Program on New Geometric Methods in Neuroscience)
Random Fractals
Fractals and Spectral Triples
Quadratic Maps: Mandelbrot and Julia Sets

Summary of Lectures

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: the logistic map, period doubling cascade, transition to chaos, Feigenbaum graphs, tent map realization of the Sharkovsky ordering, topological dimension as minimum of Hausdorff dimensions, topological properties from Hausdorff dimension, box counting dimension (lower and upper)
Tuesday January 28: discrepancies between Hausdorff, lower, and upper box counting dimensions, measures with uniform mass distribution, lower bound on Hausdorff dimension, Cantor set example, pointwise dimension and Hausdorff dimension, pointwise dimension of Bernoulli measures on shift spaces, almost everywhere limit, exact dimensionality, Shannon entropy, Khinchin axioms, Renyi entropies, Kullback-Leibler divergence, properties of Renyi entropy, escort probability, informational entropy and thermodynamical entropy, Potts model on a one-dimensional lattice
Monday February 3: box-counting Renyi dimensions, properties of Renyi dimensions, thermodynamic limits and entropy density, Renyi dimensions of Jackson Pollock paintings
Tuesday February 4: Bowen balls, local entropy, invariance under topological conjugacy, ergodic measures, Kolmogorov-Sinai entropy, topological entropy, entropy of Markov measures, Parry measures of maximal entropy, Lyapunov exponents, multifractal decompositions, Birkhoff ergodic averages, ergodic theorem
Monday February 10: consequences of ergodic theorem, laws of large numbers of Bernoulli and Markov measures, Birkhoff spectrum, maximal entropy measure, equilibrium measure, multifractal decomposition
Tuesday February 11: Menger sponge, Hausdorff dimension, topological dimension, universal property of the Menger sponge, continua and Peano continua, n-dimensional Menger spaces, universality, universality with respect to mapping, Cantor sets as projective limits of finite sets, Menger sponge and projective limits of finite graphs, category of finite graphs and connected surjections, projective limits and the category of topological graphs, topological graphs and Peano continua, Menger prespace, projective Fraisse classes, generic sequence and projective limit, Menger curve as realization, homogeneity, lifting property, universality, higher dimensional Manger compacta and inverse limits of (n-1)-connected simplicial sets, simplicial and cubical sets as functors, functorial approach to Menger spaces
Monday February 17: reading week no class
Tuesday February 18: reading week no class
Monday February 24: Continued fraction expansion, shift operator, GL2(Z) and finite index subgroups shift operator, reducible matrices semigroup, invariant measure, Gauss-Kuzmin operator and fixed point problem, transitivity property on coset of finite index subgroup
Tuesday February 25: Analytic setting for Gauss-Kuzmin operatot, Banach space, positive cone, nuclear operator, Perron-Frobenius theorem in cones in Banach spaces, shift-invariant measure and ergodicity, Schweiger fibered systems, ixmaster universe model in cosmology
Monday March 2: two-sided shift map and coding of geodesics on modular curves, geodesics on modular curve of Gamma0(2) and mixmaster solutions, moduler curves and compactification by cusps, modular forms, cusp forms of weight two and holomorphic differentials on modular curves, modular symbols and homology of modluar curves, properties of modular symbols
Tuesday March 3: limiting modular symbols, vanishing results, case of quadratic irrationalities, Lyapunov spectrum, Ruelle and Gauss-Kuzmin operators, Selberg zeta function and Fredholm determinant of the Ruelle transfer operator; Schottky groups and limit sets, Schottky groups and Jordan curves construction, limit set and domain of discontinuity, fundamental domains, Fuchsian Schottky groups
Monday March 9: Schottky groups of rank one, Banados-Teitelboim-Zanelli black hole
Tuesday March 10: random graphs, Erdos-Renyi graphs, small-world graphs, scale-free graphs, measures of centrality, phasetransitions in Erdos-Renyi graphs to one giantcomponent and to connectedness,graph Laplacian, Quantum Chaos, quantum graphs, Schroedinger equation on quantum graphs, quantum chaoson discrete graphs, spectral properties and symbolic dynamics,Ihara zeta function of graphs, poles of the Ihara zetafunction, determinant formulae,Ihara zeta function asa Ruelle zeta function, subshift of finite type and closed paths on graphs, expander graphs, spectral properties and Ihara pole distribution random matrix behavior
Monday March 16: DUE TO UNIVERSITY DIRECTIVES IN-PERSON CLASS MEETINGS SUSPENDED, more class material will be posted on this webpage
Tuesday March 17: Material for this week: continuation of Schottky groups and limit sets, Random Fractals
Monday March 23:
Tuesday March 24:
Monday March 30: student presentations (REPLACED BY IN WRITING OR REMOTE CONNECTION, contact instructor)
Tuesday March 31: student presentations (REPLACED BY IN WRITING OR REMOTE CONNECTION, contact instructor)

Some Book References

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.

Reading Materials

Links to articles and reading suggestions: (more will be added as the class progresses)

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 Feigenbaum Graphs: A Complex Network Perspective of Chaos (Luque, Lacasa, Ballestros, Robledo)
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 A Combinatorial Model for the Menger Curve (Panagiotopoulos and Solecki)
pdf Universal Menger Compacta and Universal Mappings (Dranishnikov)
pdf Characterizing k-Dimensional Universal Menger Compacta (Bestvina)
pdf Singular Homology of one-dimensional Spaces (Curtis and Fort)
pdf La dimension et la mesure (Szpilrajn)
pdf Arithmetic Quantum Chaos (Jens Marklof)
pdf Arithmetic Quantum Chaos (Peter Sarnak)
pdf What is quantum chaos? (Zeev Rudnick)
pdf Periodic Orbit Theory and Spectral Statistics for Quantum Graphs (Kottos and Smilansky)
pdf Quantum Chaos on Discrete Graphs (Smilansky)
pdf Zeta Functions and Chaos (Terras)
pdf Ergodic Theory of Simple Continued Fractions (Hines)
pdf Multifractal analysis of Lyapunov exponent for Continued fraction and Manneville-Pomeau transformations and applications to Diophantine approximation (M.Pollicott and H.Weiss)
pdf The Patterson Measure: Classics, Variantions and Applications (Denker and Stratmann)
pdf On the Bass Note of a Schottky Group (P.G. Doyle)
pdf On Boundaries of Schottky Space (H.Sato)
pdf All Fuchsian Groups are Classical Schottky Groups (J.Button)
pdf A Characterization of Schottky Groups (B.Maskit)
pdf Kleinian Schottky groups, Patterson-Sullivan measures, and Fourier decay (J.Li, F.Naud, W.Pan)
pdf On Uniformization of Riemann Surfaces and the Weil-Petersson Metric on Teichmuller and Schottky Space (P.G.Zograf, L.A.Takhtadzhyan)
pdf Schottky Space and Teichmuller Disks (F.Herrlich)
pdf Multifractal of the Apollonian tiling (Dominique Simpelaere)
pdf The multifractal analysis of Birkhoff averages and large deviations (Yakov Pesin and Howard Weiss)
pdf Weyl's Law on Riemannian Manifolds (Musser)
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)
pdf Disconnected Julia Sets (Paul Blanchard)
pdf The Mandelbrot set is universal (Curtis McMullen)