On the history of the isomorphism problem of dynamical systems with special regard to von Neumann's contribution

Details On the history of the isomorphism problem of dynamical systems with special regard to von Neumann's contribution On the history of the isomorphism problem of dynamical systems with special regard to von Neumann's contribution

2011-10-04

arxiv.org/abs/1110.0625v1

Mathematics

History and Overview

Forthcoming in: Archive for History of Exact Sciences. The final publication will be available at http://www.springer.com/ma

Scientific paper

This paper reviews some major episodes in the history of the spatial isomorphism problem of dynamical systems theory (ergodic theory). In particular, by analysing, both systematically and in historical context, a hitherto unpublished letter written in 1941 by John von Neumann to Stanislaw Ulam, this paper clarifies von Neumann's contribution to discovering the relationship between spatial isomorphism and spectral isomorphism. The main message of the paper is that von Neumann's argument described in his letter to Ulam is the very first proof that spatial isomorphism and spectral isomorphism are not equivalent because spectral isomorphism is weaker than spatial isomorphism: von Neumann shows that spectrally isomorphic ergodic dynamical systems with mixed spectra need not be spatially isomorphic.

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