Ma 4: Fractal Geometry and Chaos Theory
Spring 2011, Caltech Math Department, Tuesday-Thursday 10:30-11:55 am.
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.
Notes
- Tuesday March 29: Graphics : Logistic map, fixed points, period doubling
- Thursday, March 31: Logistic map: period three implies chaos;
- Tuesday, April 5: Linear dynamics and chaotic dynamics on the Cantor set,
coding space and coding map
- Thursday, April 7: Hausdorff measure and Hausdorff dimension
- Tuesday April 12: Fixed point theorem for contractions on
complete metric spaces
- Thursday April 14: The Hausdorff distance between compact sets,
contractions, self-similar sets
- Tuesday April 19: Topological dimension and Hausdorff dimension,
box counting dimension
- Thursday April 21: Outer measures, measures, measurable sets
- Tuesday April 26: Lebesgue measure on the line, Bernoulli measures
on Cantor sets, Banach-Tarski paradox (non-measurable sets)
- Thursday April 28: Shift spaces with Bernoulli and Markov measures
- Tuesday May 3: Measure and dimension: uniform mass principle,
local dimension, Hausdorff dimension of a measure
- Thursday May 5: Bowen balls and local entropy, topological entropy,
entropy of Markov measures
- Tuesday May 10: Lyapunov exponents, multifractals;
Shannon entropy, Khinchin axioms, Renyi entropy, information
gain, properties of Renyi entropy
- Thursday May 12: Information theoretic entropy and thermodynamical
entropy, thermodynamics formalism and statistical mechanics,
Potts models on one-dimensional lattices, coding and shift invariance
- Tuesday May 17: Renyi dimensions and box counting dimension, thermodynamicsl formalism
- Thursday May 19: Smale-Williams solenoid, hyperboli dynamical systems,
stable and unstable manifolds, attractors
- Tuesday May 24: Hausdorff dimension of products, dimension of the
solenoid attractor, Smale's horseshoe dynamical system
Students Presentation Schedule
- Tuesday May 31, 10:30-11:00, Joseph Lyndon, Percolation
- Tuesday May 31, 11:00-11:30, Gerardo Ferrara, Copulas with fractal support
- Tuesday May 31, 11:30-12:00, Karan Gupta
- Thursday June 2, 10:30-11:00 Arpit Panda, Chaos and continued fractions
- Thursday June 2, 11:00-11:30 Chris Estrada, Lyapunov exponent and continued fractions
- Thursday June 2, 11:30-12:00 Eric Zhang, Brownian Motion and Browning Surfaces
Students Presentation Topics (already assigned):
- Gerardo Ferrara:
(pdf)
G.A.Fredricks, R.B.Nelson, J.A.Rodriguez-Lallena, "Copulas
with fractal support" Insurance: Mathematics and Economics 37 (2005) 42-48
- Arpit Panda:
(pdf)
R.M.Corless, G.W.Frank, J.G.Monroe, "Chaos and continued fractions"
Physica D 46 (1990) 241-253
Additional material (links to external websites)
Textbook
Yakov Pesin and Vaughn Climenhaga, "Lectures on fractal geometry and
dynamical systems", American Mathematical Society, 2009.
Other suggested readings
- Shlomo Sternberg, "Dynamical Systems", Dover, 2010.
- Kenneth Falconer, "Fractal geometry" (2nd), Wiley, 2003.
Further suggested readings will be added as the class progresses:
Grading policy
Grading for the class will be based on participation in class
and, depending on the number of students enrolled, on
seminar presentations. The TA,
Branimir Cacic ,
will run recitation sessions.
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