Ma 4: Fractal Geometry and Chaos Theory
Spring 2017, Caltech Math Department, Tuesday-Thursday 1:00-2:25 pm,
Room SLN 257 (TA: Josh Lieber).
Instructor:
Matilde Marcolli
Brief Course Description
The class will give an introduction to the geometry of fractals and
to their occurrence in the context of dynamical systems and in relation
to chaos theory.
Bibliography
- Yakov Pesin and Vaughn Climenhaga, "Lectures on fractal geometry and
dynamical systems", American Mathematical Society, 2009.
- Shlomo Sternberg, "Dynamical Systems", Dover, 2010.
- 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.
- Kenneth Falconer, "Fractal geometry" (2nd), Wiley, 2003.
Other suggested readings
Additional reading material will be posted here as the
class progresses
- pdf
Period three implies chaos (by T.Y.Li and A.Yorke)
- 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)
Lectures Outlines
Notes will be posted here.
- Tuesday April 4: The Lorenz equations: fixed points and
linearization, bifurcation, stability, Lyapunov function,
attractor sets, chaotic regime; sensitive dependence on initial conditions
and chaotic dynamics, strange attractors and fractals
- Thursday April 6: Discrete dynamical systems, iteration and orbits,
piecewise linear functions and domains of iteration, construction of
the Cantor set, coding of the dynamics via shift map, fixed points and
periodic orbits, general topological properties
- Tuesday April 11: Geometry and topology of the Cantor set,
metric, cylinder sets, coverings and the notion of topological dimension,
Hausdorff dimension, fractals
- Thursday April 13: Hausdorff measures and Hausdorff dimension,
computation of the Hausdorff dimension of the Cantor set, non-uniform
Cantor sets, Lipschitz functions and Hausdorff dimension.
- Tuesday April 18: Topological dimension as minimum of Hausdorff dimension
- Thursday April 20: Box counting dimension and relation to Hausdorff dimension, sets with zero Hausdorff dimension and non-zero box counting dimension
- Tuesday April 18: Fixed point theorem for contractions on complete
metric spaces, Hausdorff distance on non-empty compact sets, completeness, iterated function systems and self-similar sets, self-similarity (scaling)
dimension, Sierpinski gaskets and non-uniform Cantor sets
- Thursday April 20: sigma-algebras and measures, outer measures,
non-measurable sets, Bernoulli measures on Cantor sets, Bernoulli and
Hausdorff measure, shift-invariance
- Tuesday April 25: the Banach-Tarski paradox (duplication of the sphere, nonmeasurable sets and the axiom of choice), Markov measures on Cantor sets,
support of measures, subshifts of finite type, measures and pointwise dimension
- Thursday April 27: pointwise dimension and Hausdorff dimension,
pointwise dimension of Bernoulli measures, almost everywhere value, dense
bad set of measure zero
- Tuesday May 2: ergodic measures, ergodicity on Cantor sets
- Thursday May 4: Entropy for dynamical systems, Kolomogorov-Sinai
entropy, topological entropy, entropy and dimension
- Tuesday May 9: The Shannon entropy: Khinchin axioms and axiomatic
characterization
- Thursday May 11: Renyi entropy: properties and estimates
- Tuesday May 16: Renyi dimensions and dynamical systems
- Thursday May 18: the Williams solenoid chaotic dynamical system,
attractor sets and dynamics on attractor sets
- Tuesday May 23: the Smale horseshoe: attractor set, Hausdorff
dimension, dynamics on the attractor set, coding by invertible subshift
of finite type
- Thursday May 25: the Lorenz attractor revisited: Poincare' map
and shorseshoe dynamics
- Tuesday May 30: student presentations
- Thursday June 1: student presentations
Grading policy
A grade for the class will be assigned on the basis
of participation in class and of an oral presentation
based on assigned reading material.
Schedule of Student Presentations
A schedule of student presentations will be added here
- Tuesday May 30: Allison Nicole Penn, TBA
- Tuesday May 30: Erik Michael Urzua, "Period three implies Chaos"
- Thursday June 1: Koichiro Kajikawa,
"Architectural Bias in Recurrent Neural Networks: Fractal Analysis"
- Thursday June 1: Brian Lee, "Arithmetic Quantum Chaos"
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