Learning objectives
To give fundamental ideas and methods in studying stable and instable systems
Prerequisites
Basic courses in analysis and algebra of a first degree in physics
Course unit content
Introduction to topological dynamics and ergodic theory with applications
Full programme
Preliminary notions of topology and measure theory
One parameter groups, flows and maps.
Elementary examples of one-dimensional maps (circle, quadratic family etc.)
Attractors. Iperbolicity.
Topological chaos.
Shift spaces and symbolical dynamics.
Topological conjugacy.
Structural stability.
Probability spaces. Metric dynamical systems.
Invariant measures.
Birkhoff’s Theorem.
Ergodicity and mixing.
Metric isomorphism.
Shannon’s entropy and Khinchin Theorem.
Kolmogorov’s Entropy and Topological Entropy.
Bernoulli Shifts.
Predictability and chaos.
Cellular Automata, examples (Ising, sandpiles etc.)
Analysis of temporal series, Wiener-Khinchin Theorem.
Further possible arguments:
Insights on the hamiltonian case.
Reading of a review paper
Bibliography
R.L.Devaney: Chaotic Dynamical Systems (Benjamin 1985);
A.J. Lichtenberg and M.A. Liebermann: Regular and Stochastic Motion (Springer 1983);
V.I.Arnold and A. Avez: Ergodic Problems of Classical Mechanics (Benjamin 1968)
D. Ruelle: Chaotic Evolution and Strange Attractors (Cambridge UP 1989)
R. Badii and A. Politi: Complexity (Cambridge UP 1997)
T. Toffoli and N. Margulis: Cellular Automata Machines (Mit Press 1987)
A.Vulpiani: Determinismo e Caos (Carocci 2004)
P.Castiglione. M. Falcioni, A. Lesne, A. Vulpiani: Chaos and Corse Graining in Statistical Mechanics (Cambridge UP, 2008)
Teaching methods
Frontal lessons with exercises
Assessment methods and criteria
Personal research and presentation of an argument regarding the content of the course.
Other information
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