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Ergodic theory

In a series of papers published in 1932, von Neumann made foundational contributions to ergodic theory, a branch of mathematics that involves the states of dynamical systems with an invariant measure. Of the 1932 papers on ergodic theory, Paul Halmos wrote that even "if von Neumann had never done anything else, they would have been sufficient to guarantee him mathematical immortality". By then von Neumann had already written his articles on operator theory, and the application of this work was instrumental in his mean ergodic theorem.

The theorem is about arbitrary one-parameter unitary groups t → V t {\displaystyle {\mathit {t}}\to {\mathit {V_{t}}}} and states that for every vector ϕ {\displaystyle \phi } in the Hilbert space, lim T → ∞ 1 T ∫ 0 T V t (ϕ) d t {\textstyle \lim _{T\to \infty }{\frac {1}{T}}\int _{0}^{T}V_{t}(\phi)\,dt} exists in the sense of the metric defined by the Hilbert norm and is a vector ψ {\displaystyle \psi } which is such that V t (ψ) = ψ {\displaystyle V_{t}(\psi)=\psi } for all t {\displaystyle t}. This was proven in the first paper. In the second paper, von Neumann argued that his results here were sufficient for physical applications relating to Boltzmann's ergodic hypothesis. He also pointed out that ergodicity had not yet been achieved and isolated this for future work.

Later in the year he published another influential paper that began the systematic study of ergodicity. He gave and proved a decomposition theorem showing that the ergodic measure preserving actions of the real line are the fundamental building blocks from which all measure preserving actions can be built. Several other key theorems are given and proven. The results in this paper and another in conjunction with Paul Halmos have significant applications in other areas of mathematics.