Editing John von Neumann (section)

=== 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]].<ref>{{cite journal|author-link=Eberhard Hopf|first=Eberhard|last=Hopf|title=Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung|year=1939|journal=Leipzig Ber. Verhandl. Sächs. Akad. Wiss.|volume=91|pages=261–304|language=German}} {{pb}}
Two of the papers are: {{pb}}
{{cite journal|first=John|last=von Neumann|title=Proof of the Quasi-ergodic Hypothesis|year=1932|journal=Proc Natl Acad Sci USA|volume=18|pages=70–82|doi=10.1073/pnas.18.1.70|pmid=16577432|issue=1|pmc=1076162|bibcode=1932PNAS...18...70N |doi-access=free |bibcode-access=free }} {{pb}}
{{cite journal|first=John|last=von Neumann|title=Physical Applications of the Ergodic Hypothesis|year=1932|journal=Proc Natl Acad Sci USA|volume=18|pages=263–266|doi=10.1073/pnas.18.3.263|pmid=16587674|issue=3|pmc=1076204|jstor=86260|bibcode=1932PNAS...18..263N|doi-access=free}}.</ref> 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".{{sfn|Halmos|1958|p=93}} By then von Neumann had already written his articles on [[operator theory]], and the application of this work was instrumental in his [[Ergodic theory#Mean ergodic theorem|mean ergodic theorem]].{{sfn|Halmos|1958|p=91}}

The theorem is about arbitrary [[One-parameter group|one-parameter]] [[unitary group]]s <math>\mathit{t} \to \mathit{V_t}</math> and states that for every vector <math>\phi</math> in the [[Hilbert space]], <math display=inline>\lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} V_t(\phi) \, dt</math> exists in the sense of the metric defined by the Hilbert norm and is a vector <math>\psi</math> which is such that <math>V_t(\psi) = \psi</math> for all <math>t</math>. 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 [[Ludwig Boltzmann|Boltzmann's]] [[ergodic hypothesis]]. He also pointed out that [[ergodicity]] had not yet been achieved and isolated this for future work.<ref name=mackey1990>{{harvc |last1=Mackey |first1=George W. |author-link=George Mackey |year=1990 |chapter=Von Neumann and the Early Days of Ergodic Theory |in1=Glimm |in2=Impagliazzo |in3=Singer |pages=27–30}}</ref>

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 dynamical system|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.{{r|mackey1990}}<ref>{{harvc |last1=Ornstein |first1=Donald S. |author-link=Donald Samuel Ornstein |year=1990 |chapter=Von Neumann and Ergodic Theory |in1=Glimm |in2=Impagliazzo |in3=Singer |page=39}}</ref>