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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 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.

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 Lt defines a flow on R preserving m and hence a flow λt on L∞(R). Let S = L1 with corresponding automorphism σ of L∞(R). Thus τ ⊗ σ gives an automorphism of A L∞(R) which commutes with the flow id ⊗ λt. 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 ⊗ λt to B. Since λt 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.

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 Rλ(t)R∗ = Vt where λ(t) is translation by t and Vt is multiplication by eitx. Thus Vt lies in L∞(R). In particular V1 = R S R∗. A ceiling function h is a function in A with h ≥ ε1 with ε > 0. Then eihx 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 ⊗Vt. So 1=W1 = (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 commuting with 1=T ⊗ S. The induced flow σt is given by 1= σt (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 commuting with 1=(T ⊗ I) W1. The induced flow corresponds to the ceiling function h ≡ 1, the constant function. Again W1, and hence 1=(T ⊗ I) W1, commutes with I ⊗ λ(t). The special flow on B(H) is again given by 1=σt (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.

If St is an ergodic flow on the measure space (X,μ) corresponding to a 1 parameter group of automorphisms σt of A = L∞(X,μ), then by the Hopf decomposition either every St with t ≠ 0 is dissipative or every St 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 σt(p) < p for t > 0 and λ(p – σt(p)) = t. In this case σt(p) =E([t,∞)) where E is a projection valued measure on R. These projections generate a von Neumann subalgebra B of A. By ergodicity σt(p) 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 St, it is implemented by a unitary representation Ut. By the Stone–von Neumann theorem for the covariant system B, Ut, the Hilbert space H = L2(X,λ) admits a decomposition L2(R) ⊗ where B and Ut 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 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 σt(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 and is not a multiple of 1 ⊗ I. But such an element commutes with Ut so is fixed by σt, contradicting ergodicity. Hence A = B = L∞(R). When all the σt 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 ≤ σt (p) ∨ σ2t (p) ∨ σ3t (p) ∨ ⋅⋅⋅ In particular ∨±t>0 σt (p) = 1 for p ≠ 0.