In Portuguese, these notes compile the topics covered in the course “Introduction to Dynamical Systems” that has be taught at UFC in the last years.
The purpose of these lecture notes was to help the author in the organization of the mini-course “Introduction to Ratner theory”, held at IMPA on August 2013, as part of the activities of the thematic semester “Dynamics Beyond Uniform Hyperbolicity”.
The aim of the mini-course was to introduce Ph.D. students to some beautiful topics of dynamics on homogeneous spaces, with emphasis on Ratner’s theorems on the classification of orbit closures, invariant measures and equidistribution of actions of subgroups generated by unipotent elements on homogeneous spaces.
The Szemeréd regularity lemma is an important tool in discrete mathematics, specially in graph theory and additive combinatorics. The lemma can be used to theorems for arbitrary graphs whenever the corresponding result is easy for random graphs. One example is the graph removal lemma, which can be used to prove Szemerédi’s theorem on the existence of arithmetic progressions.
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