Ma4/104c Spring 2026: Fractal Geometry and Chaos Theory
Caltech, Linde Hall Room 387, Tuesday-Thursday 10:30-11:55 am
Instructor:
Matilde Marcolli
This class will cover various aspects of fractal geometry and its
relation to chaos theory. 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 spaces. Students will give a final presentation
on reading material chosen from a list posted to this webpage.
Slides of Lectures
Slides of lectures will be posted here as the class progresses
Summary of lectures
- Tuesday March 31:
- Thursday April 2:
- Tuesday April 7:
- Thursday April 9:
- Tuesday April 14:
- Thursday April 16:
- Tuesday April 21:
- Thursday April 23:
- Tuesday April 28:
- Thursday April 30:
- Tuesday May 5:
- Thursday May 7:
- Tuesday May 12:
- Thursday May 14:
- Tuesday May 19: cancelled due to travel
- Thursday May 21:
- Tuesday May 26:
- Thursday May 28:
- Friday May 29: students presentations (room and schedule to be announced)
Reading Materials
There is no specific textbook for the class. Some suggested
reading material will be posted here.
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)
Schedule of Presentations