Ma 4: Fractal Geometry and Chaos Theory
Spring 2014, Caltech Math Department, Tuesday-Thursday 10:30-11:55 am,
Room SLN 257.
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
- Tuesday April 1: Lorentz attractor: fixed points, linearization,
Lyapunov function,
pitchfork bifurcation, Hopf bifurcation, strange attractor; sensitive
dependence on the initial conditions, Lyapunov exponent;
definition of chaos
- Thursday April 3: Discrete dynamical systems, iterates, domains,
cobweb diagrams, construction of the Cantor sets via domains of iterates
of a piecewise linear map, coding space for the Cantor set
- Tuesday April 8: Topological spaces and metric spaces, cylinder
sets and topology on the Cantor set, non-uniform Cantor sets
- Thursday April 10: Hausdorff measures and Hausdorff dimension,
jumping infinity/zero, scaling behavior, estimate of the Hausdorff
dimension of the Cantor set; scaling transformations and Lipschitz functions
- Tuesday April 15: guest lecture by Jan-Jitse Venselaar: Dirac
operators and spectral triples on the circle and on tori
- Thursday April 17: guest lecture by Jan-Jitse Venselaar:
a Dirac operator on the Cantor set, spectral triple, zeta function
and the Hausdorff dimension
- Tuesday April 22: Fixed point theorem for contractions on
complete metric spaces; the Hausdorff distance on nonempty compact
subsets of a complete metric space; completeness of the Hausdorff
distance; contraction maps in the Hausdorff distance and self-similar
sets; the Cantor set and the Sierpinski gasket as self-similar sets
- Thursday April 24: topological dimension and relation to the
Hausdorff dimension, Lipschitz and bi-Lipschitz functions and the
Hausdorff dimension, topological consequences of small Hausdorff dimension
(total disconnectedness)
- Tuesday April 29: box counting dimension and relation to the
Hausdorff dimension, properties of box counting, examples with zero
Hausdorff dimension and arbitrary lower and upper box counting
dimensions between zero and one
- Thursday May 1: Measure theory: outer measures, sigma-algebras,
measures from outer measures, Caratheodory construction, Dirac delta
measure, counting measure, Lebesgue and Hausdorff measures, non-measurable
sets: Banach-Tarsky paradox
- Tuesday May 6: Bernoulli probability measures on shift spaces, Markov
measures on shift spaces, support of a measure, admissible words,
admissibility matrix and subshifts of finite type
- Thursday May 8: Uniform mass distribution principle, lower bounds
on the Hausdorff dimension from measures, local dimension, case of
Cantor sets, relation to Shannon entropy, almost everywhere convergence
of digit frequencies; ergodic measures, ergodicity of Bernoulli measures
- Tuesday May 13: Bowen balls, local entropy, invariance of entropy
under topological conjugacy, topological entropy, entropy of Markov measures,
Parry measures of maximal entropy and Perron-Frobenius theory of
the admissibility matrix
- Thursday May 15: no class today: Ditch Day!
- Tuesday May 20:
- Thursday May 22:
- Tuesday May 27: Student presentations
- Thursday May 29: Student presentations
- Tuesay June 3: Student presentations
- Thursday June 5: 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
- May 27, Sadaf Amouzegar: Approximations of continuous Newton's method
- May 27, Hugo Lavenant: Fifty years of entropy in dynamics
- May 29, Evan Patterson: The multifractal analysis of Birkhoff averages
and large deviations
- May 29, Zachary Rivchin
- June 3, Melissa Zhang: Complex dimension of self-similar fractal
strings and Diophantine approximations
- June 3, Taylor Sturmwasser: Period Three implies Chaos
- June 5, Anusha Sinha: What is Quantum Chaos?
- June 5, Daniel Guth: Spectral triples and the geometry of fractals
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