Ergodic theory deals with the study of invariant measures in dynamical systems. "Measures" are generalizations of concepts such as length, area, volume, etc. "Invariant measures" are measures that are preserved by functions. "Dynamical systems" are systems in which a fixed rule governs the time dependence of points in a geometric space, such as the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish in a lake each spring.

An introductory class in ergodic theory is likely to include the following topics:

  • Measure-preserving transformations
  • Isomorphism, conjugacy, and spectral isomorphism
  • Measure-preserving transformations with discrete spectrum
  • Entropy
  • Topological dynamics
  • Invariant measures for continuous transformations
  • Topological entropy
  • Relationship between topological entropy and measure-theoretic entropy
  • Topological pressure and its relationship with invariant measures

There is no lack of books on this interesting mathematical topic, some of which can be seen on or Google. In addition, students should spend time with the journal of Ergodic Theory and Dynamical Systems, put out by Cambridge Journals Online.

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