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In mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been understood in terms of the theory of unitary representations of locally compact groups: if Γ is the fundamental group of a closed surface, regarded as a discrete subgroup of the Möbius group G = PSL(2,R), then the geodesic and horocycle flow can be identified with the natural actions of the subgroups A of real positive diagonal matrices and N of lower unitriangular matrices on the unit tangent bundle G / Γ. The Ambrose-Kakutani theorem expresses every ergodic flow as the flow built from an invertible ergodic transformation on a measure space using a ceiling function. In the case of geodesic flow, the ergodic transformation can be understo

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rdfs:label	Ergodic flow (en)
rdfs:comment	In mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been understood in terms of the theory of unitary representations of locally compact groups: if Γ is the fundamental group of a closed surface, regarded as a discrete subgroup of the Möbius group G = PSL(2,R), then the geodesic and horocycle flow can be identified with the natural actions of the subgroups A of real positive diagonal matrices and N of lower unitriangular matrices on the unit tangent bundle G / Γ. The Ambrose-Kakutani theorem expresses every ergodic flow as the flow built from an invertible ergodic transformation on a measure space using a ceiling function. In the case of geodesic flow, the ergodic transformation can be understo (en)
dcterms:subject	Ergodic theory
Wikipage page ID	2781451 (xsd:integer)
Wikipage revision ID	1050218023 (xsd:integer)
Link from a Wikipage to another Wikipage	Homogeneous space Hopf decomposition Riemann surface Von Neumann algebra Ann. of Math. Math. Ann. Mathematics Gelfand representation Tangent bundle Classical mechanics Geodesic Geometry Fundamental group Closed surface Horocycle Spectrum of a C*-algebra Stone's theorem on one-parameter unitary groups Measure space Locally compact group American Mathematical Society Ergodic theory Symbolic dynamics Ergodic theory Chebyshev's inequality Axiom A Institute for Advanced Study Bruhat decomposition Mutatis mutandis Strong operator topology Unitary representation University of Chicago Press Weak operator topology Möbius group Radon−Nikodym derivative Hyperbolic surface Proceedings of Symposia in Pure Mathematics Geodesic flow Gustav Hedlund Anosov flow
Link from a Wikipage to an external page	https://archive.org/details/Olden1/mode/2up https://play.google.com/books/reader%3Fid=q0ulXFdcpK0C&printsec=frontcover&pg=GBS.PA171
sameAs	Ergodic flow Ergodic flow
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Harvtxt dbt:Main dbt:Math dbt:Overline dbt:Reflist
has abstract	In mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been understood in terms of the theory of unitary representations of locally compact groups: if Γ is the fundamental group of a closed surface, regarded as a discrete subgroup of the Möbius group G = PSL(2,R), then the geodesic and horocycle flow can be identified with the natural actions of the subgroups A of real positive diagonal matrices and N of lower unitriangular matrices on the unit tangent bundle G / Γ. The Ambrose-Kakutani theorem expresses every ergodic flow as the flow built from an invertible ergodic transformation on a measure space using a ceiling function. In the case of geodesic flow, the ergodic transformation can be understood in terms of symbolic dynamics; and in terms of the ergodic actions of Γ on the boundary S1 = G / AN and G / A = S1 × S1 \ diag S1. Ergodic flows also arise naturally as invariants in the classification of von Neumann algebras: the flow of weights for a factor of type III0 is an ergodic flow on a measure space. (en)
prov:wasDerivedFrom	wikipedia-en:Ergodic_flow?oldid=1050218023&ns=0
page length (characters) of wiki page	36436 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Ergodic_flow
is Link from a Wikipage to another Wikipage of	Anosov diffeomorphism Hopf decomposition Conductance (graph) Crossed product Jordan operator algebra Axiom A Ambrose−Kakutani theorem
is Wikipage redirect of	Ambrose−Kakutani theorem
is foaf:primaryTopic of	wikipedia-en:Ergodic_flow

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