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 "S"1 = "G" / "AN" and "G" / "A" = "S"1 × "S"1 \ diag "S"1. 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.
Hedlund's theorem: ergodicity of geodesic and horocycle flows.
The method using representation theory relies on the following two results:
(1) As a topological space, the homogeneous space "X" = "G" / "N" can be identified with R2 \ {0} with the standard action of "G" as 2 × 2 matrices. The subgroup of "N" has two kinds of orbits: orbits parallel to the "x"-axis with "y" ≠ 0; and points on the "x"-axis. A continuous function on "X" that is constant on "N"-orbits must therefore be constant on the real axis with the origin removed. Thus the matrix coefficient ψ("x") = ("x"ξ,ξ) satisfies ψ("g") = 1 for "g" in "A" · "N". By unitarity, ||"g"ξ − ξ||2 = 2 − ψ("g") − ψ("g"–1) = 0, so that "g"ξ = ξ for all "g" in "B" = "A" · "N" = "N" · "A". Now let "s" be the matrix formula_1. Then, as is easily verified, the double coset "BsB" is dense in "G"; this is a special case of the Bruhat decomposition. Since ξ is fixed by "B", the matrix coefficient ψ("g") is constant on "BsB". By density, ψ("g") = 1 for all "g" in "G". The same argument as above shows that "g"ξ = ξ for all "g" in "G".
(2) Suppose that ξ is fixed by "A". For the unitary 1-parameter group "N" ≅ R, let "P"["a","b"] be the spectral subspace corresponding to the interval ["a","b"]. Let "g"("s") be the diagonal matrix with entries "s" and "s"−1 for |"s"| ; 1. Then "g"("s")"P"["a","b"]"g"("s")−1 = "P"["s"2"a", "s"2"a"]. As |"s"| tends to infinity the latter projections tend to 0 in the strong operator topology if 0 ; "a" ; "b" or "a" ; "b" ; 0. Since "g"("s")ξ = ξ, it follows "P"["a","b"]ξ = 0 in either case. By the spectral theorem, it follows that ξ is in the spectral subspace "P"({0}); in other words ξ is fixed by "N". But then, by the first result, ξ must be fixed by "G".
The classical theorems of Gustav Hedlund from the early 1930s assert the ergodicity of the geodesic and horocycle flows corresponding to compact Riemann surfaces of constant negative curvature. Hedlund's theorem can be re-interpreted in terms of unitary representations of "G" and its subgroups. Let Γ be a cocompact subgroup of PSL(2,R) = "G" / {±"I"} for which all non-scalar elements are hyperbolic. Let "X" = Γ \ "G" / "K" where "K" is the subgroup of rotations formula_2. The unit tangent bundle is "SX" = Γ \ "G", with the geodesic flow given by the right action of "A" and the horocycle flow by the right action of "N". This action if ergodic if 
L∞(Γ \ "G")"A" = C, i.e. the functions fixed by "A" are just the constant functions. Since Γ \ "G" is compact, this will be the case ifL2(Γ \ "G")"A" = C. Let "H" = L2(Γ \ "G"). Thus "G" acts unitarily on "H" on the right. Any non-zero ξ in "H" fixed by "A" must be fixed by "G", by the second result above. But in this case, if "f" is a continuous function on "G" of compact support with ∫ "f" = 1, then ξ = ∫ "f"("g") "g"ξ "dg". The right hand side equals ξ ∗ "f", a continuous function on "G". Since ξ is right-invariant under "G", it follows that ξ is constant, as required. Hence the geodesic flow is ergodic. Replacing "A" by "N" and using the first result above, the same argument shows that the horocycle flow is ergodic.
Ambrose−Kakutani–Krengel–Kubo theorem.
Induced flows.
Examples of flows induced from non-singular invertible transformations of measure spaces were defined by in his operator-theoretic approach to classical mechanics and ergodic theory. Let "T" be a non-singular invertible transformation of ("X",μ) giving rise to an automorphism τ of "A" = L∞("X"). This gives rise to an invertible transformation "T" ⊗ id of the measure space ("X" × R,μ × "m"), where "m" is Lebesgue measure, and hence an automorphism τ ⊗ id of A ⊗ L∞(R). Translation "L""t" defines a flow on R preserving "m" and hence a flow λ"t" on L∞(R). Let "S" = "L"1 with corresponding automorphism σ of L∞(R). Thus τ ⊗ σ gives an automorphism of "A" ⊗ L∞(R) which commutes with the flow id ⊗ λ"t". The induced measure space "Y" is defined by "B" = L∞("Y") = L∞("X" × R)τ ⊗ σ, the functions fixed by the automorphism τ ⊗ σ. It admits the "induced flow" given by the restriction of id ⊗ λ"t" to "B". Since λ"t" acts ergodically on L∞(R), it follows that the functions fixed by the flow can be identified with L∞("X")τ. In particular if the original transformation is ergodic, the flow that it induces is also ergodic.
Flows built under a ceiling function.
The induced action can also be described in terms of unitary operators and it is this approach which clarifies the generalisation to special flows, i.e. flows built under ceiling functions. Let "R" be the Fourier transform on L2(R,"m"), a unitary operator such that "R"λ("t")"R"∗ = "V""t" where λ("t") is translation by "t" and "V""t" is multiplication by e"itx". Thus "V""t" lies in L∞(R). In particular "V"1 = "R" "S" "R"∗. A ceiling function "h" is a function in "A" with "h" ≥ ε1 with ε ; 0. Then e"ihx" gives a unitary representation of R in "A", continuous in the strong operator topology and hence a unitary element "W" of A ⊗ L∞(R), acting on L2("X",μ) ⊗ L2(R). In particular "W" commutes with "I" ⊗"V""t". So "W"1 = ("I" ⊗ "R"∗) "W" ("I" ⊗ "R") commutes with "I" ⊗ λ("t"). The action "T" on L∞("X") induces a unitary "U" on L2("X") using the square root of the Radon−Nikodym derivative of μ ∘ "T" with respect to μ. The induced algebra "B" is defined as the subalgebra of "A" ⊗ L∞(R) commuting with "T" ⊗ "S". The induced flow σ"t" is given by σ"t" ("b") = ("I" ⊗ λ("t")) "b" ("I" ⊗ λ(−"t")).
The "special flow corresponding to the ceiling function" "h" "with base transformation" "T" is defined on the algebra "B"("H") given by the elements in "A" ⊗ L∞(R) commuting with ("T" ⊗ "I") "W"1. The induced flow corresponds to the ceiling function "h" ≡ 1, the constant function. Again "W"1, and hence ("T" ⊗ "I") "W"1, commutes with "I" ⊗ λ("t"). The special flow on "B"("H") is again given by σ"t" ("b") = ("I" ⊗ λ("t")) "b" ("I" ⊗ λ(−"t")). The same reasoning as for induced actions shows that the functions fixed by the flow correspond to the functions in "A" fixed by σ, so that the special flow is ergodic if the original non-singular transformation "T" is ergodic.
Relation to Hopf decomposition.
If "S""t" is an ergodic flow on the measure space ("X",μ) corresponding to a 1-parameter group of automorphisms σ"t" of "A" = L∞("X",μ), then by the Hopf decomposition either every "S""t" with "t" ≠ 0 is dissipative or every "S""t" with "t" ≠ 0 is conservative. In the dissipative case, the ergodic flow must be transitive, so that "A" can be identified with L∞(R) under Lebesgue measure and R acting by translation.
To prove the result on the dissipative case, note that "A" = L∞("X",μ) is a maximal Abelian von Neumann algebra acting on the Hilbert space L2("X",μ). The probability measure μ can be replaced by an equivalent invariant measure λ and there is a projection "p" in "A" such that σ"t"("p") ; "p" for "t" ; 0 and λ("p" – σ"t"("p")) = "t". In this case σ"t"("p") ="E"(["t",∞)) where "E" is a projection-valued measure on R. These projections generate a von Neumann subalgebra "B" of "A". By ergodicity σ"t"("p") formula_3 1 as "t" tends to −∞. The Hilbert space L2("X",λ) can be identified with the completion of the subspace of "f" in "A" with λ(|"f"|2) ; ∞. The subspace corresponding to "B" can be identified with L2(R) and "B" with L∞(R). Since λ is invariant under "S""t", it is implemented by a unitary representation "U""t". By the Stone–von Neumann theorem for the covariant system "B", "U""t", the Hilbert space "H" = L2("X",λ) admits a decomposition L2("R") ⊗ formula_4 where "B" and "U""t" act only on the first tensor factor. If there is an element "a" of "A" not in "B", then it lies in the commutant of "B" ⊗ C, i.e. in "B" ⊗ B(formula_4). If can thus be realised as a matrix with entries in "B". Multiplying by χ["r","s"] in "B", the entries of "a" can be taken to be in L∞(R) ∩ L1(R). For such functions "f", as an elementary case of the ergodic theorem the average of σ"t"("f") over [−"R","R"] tends in the weak operator topology to ∫ "f"("t") "dt". Hence for appropriate χ["r","s"] this will produce an element in "A" which lies in C ⊗ B(formula_4) and is not a multiple of 1 ⊗ "I". But such an element commutes with "U""t" so is fixed by σ"t", contradicting ergodicity. Hence "A" = "B" = L∞(R).
When all the σ"t" with "t" ≠ 0 are conservative, the flow is said to be "properly ergodic". In this case it follows that for every non-zero "p" in "A" and "t" ≠ 0, "p" ≤ σ"t" ("p") ∨ σ2"t" ("p") ∨ σ3"t" ("p") ∨ ⋅⋅⋅ In particular ∨±"t" ;0 σ"t" ("p") = 1 for "p" ≠ 0.
Theorem of Ambrose–Kakutani–Krengel–Kubo.
The theorem states that every ergodic flow is isomorphic to a special flow corresponding to a ceiling function with ergodic base transformation. If the flow leaves a probability measure invariant, the same is true of the base transformation.
For simplicity only the original result of is considered, the case of an ergodic flow preserving a probability measure μ. Let "A" = L∞("X",μ) and let σ"t" be the ergodic flow. Since the flow is conservative, for any projection "p" ≠ 0, 1 in "A" there is a "T" ; 0 without σ"T"("p") ≤ "p", so that (1 − "p") ∧ σ"T"("p") ≠ 0. On the other hand, as "r" ; 0 decreases to zero
formula_7
in the strong operator topology or equivalently the weak operator topology (these topologies coincide on unitaries, hence involutions, hence projections). Indeed, it suffices to show that if ν is any finite measure on "A", then ν("a""r") tends to ν("p"). This follows because "f"("t") = ν(σ"t"("p")) is a continuous function of "t" so that the average of "f" over [0,"r"] tends to "f"(0) as "r" tends to 0.
Note that 0 ≤ "a""r" ≤ 1. Now for fixed "r" ; 0, following , set
formula_8
Set "r" = "N"–1 for "N" large and "f""N" = "a""r". Thus 0 ≤ "f""N" ≤ 1 in L∞("X",μ) and "f""N" tends to a characteristic function "p" in L1("X",μ). But then, if ε = 1/4, it follows that χ[0,ε]("f""N") tends to χ[0,ε]("p") = 1 – "p" in L1("X"). Using the splitting "A" = "pA" ⊕ (1 − "p")"A", this reduces to proving that if 0 ≤ "h""N" ≤ 1 in L∞("Y",ν) and "h""N" tends to 0 in L1("Y",ν), then χ[1−ε,1]("h""N") tends to 0 in L1("Y",ν). But this follows easily by Chebyshev's inequality: indeed (1−ε) χ[1−ε,1]("h""N") ≤ "h""N", so that ν(χ[1−ε,1]("h""N")) ≤