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                {\Large \bf Dynamical Systems}\\[.2in]
{Math 675 - Fall 98}\\[.2in]
		{\large \bf Viorel Nitica}\\[.2in]
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The goal of this course
is to give an {\it introductory} exposition to the theory of dynamical
systems.
This course might be of interest to graduate students
and researchers in differential geometry, topology and complex dynamics.

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The theory of dynamical systems is a major mathematical discipline that
is intertwined with all the major areas of mathematics: analysis, number
theory, differential geometry, topology, probability. Ideas coming from
dynamical
systems stimulate research in many sciences, and give rise to the vast new
area called nonlinear dynamics, or chaos theory.

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The course will start with a discussion of the following elementary
examples: rotations of the circle, translations of the torus,
linear flows on the torus, expanding
maps, hyperbolic toral automorphisms, geodesic and horocycle flows
on surfaces of constant
negative curvature, and symbolic dynamical systems.

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These examples are used to formulate a general program aimed toward the study
of the asymptotic properties (minimality, topological transitivity, growth
of orbits,
ergodicity, mixing) of dynamical systems, as well as to introduce
the principal invariants (differential and topological equivalence, moduli,
asymptotic orbit growth, entropy, cocycles and cohomological
equations), and several important methods (fixed point methods, coding,
KAM-type Newton method, local normal forms).

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The last part of the course will be dedicated to the study of the hyperbolic
dynamical systems: existence of the stable and unstable manifolds, closing
lemma, shadowing of pseudo-orbits, stability of hyperbolic sets, Markov
partitions, local product structure, ergodicity
of geodesic flows on compact Riemannian manifolds with negative curvature,
Livsic theorem, SRB measures).

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Main reference: Introduction to the modern theory of dynamical systems, by
A. Katok and B. Hasselblatt.

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