- Understanding basic concepts and theorems in ergodic theory, such as ergodicity, mixing properties, and recurrence.
- Applying measure-theoretic techniques to study dynamical systems, including probability theory and Lebesgue integration.
- Analyzing the behavior of dynamical systems under various transformations, such as shifts, rotations, and translations.
- Using tools from functional analysis, harmonic analysis, and probability theory to analyze the dynamics of systems.
- Developing mathematical reasoning and problem-solving skills through the study of real-world applications of ergodic theory.
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Fundamentals of measure theory (measure spaces, measurable functions, the Lebesgue measure).
Ergodic transformations (definition of ergodic transformations, invariant measures, the ergodic decomposition theorem).
Ergodic theorems ( mean ergodic theorem, Birkhoff's ergodic theorem and convergence in measure).
Mixing properties (dynamic systems, weak mixing, strong mixing, topological mixing).
Applications of ergodic theory (to number theory, statistical mechanics, and probability theory).
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