Measure theory | Ergodic theory

Hopf decomposition

In mathematics, the Hopf decomposition, named after Eberhard Hopf, gives a canonical decomposition of a measure space (X, μ) with respect to an invertible non-singular transformation T:X→X, i.e. a transformation which with its inverse is measurable and carries null sets onto null sets. Up to null sets, X can be written as a disjoint union C ∐ D of T-invariant sets where the action of T on C is conservative and the action of T on D is dissipative. Thus, if τ is the automorphism of A = L∞(X) induced by T, there is a unique τ-invariant projection p in A such that pA is conservative and (I–p)A is dissipative. (Wikipedia).

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Hopf Fibration 1

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From playlist 3D printing

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From playlist 3D printing

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From playlist Summer of Math Exposition Youtube Videos

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From playlist Research Abstracts from Brunton Lab

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From playlist Ready

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From playlist Quantum Groups Seminar

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Related pages

Conservative system | Mathematics | Ergodic flow | Measure space | Wandering set | Null set