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.
Hedlund's theorem: ergodicity of geodesic and horocycle flows The method using representation theory relies on the following two results:
If = acts unitarily on a Hilbert space and is a unit vector fixed by the subgroup of upper unitriangular matrices, then is fixed by . If = acts unitarily on a Hilbert space and is a unit vector fixed by the subgroup of diagonal matrices of determinant , then is fixed by .
(1) As a topological space, the homogeneous space = can be identified with } with the standard action of as matrices. The subgroup of has two kinds of orbits: orbits parallel to the -axis with ; and points on the -axis. A continuous function on that is constant on -orbits must therefore be constant on the real axis with the origin removed. Thus the matrix coefficient = satisfies = for in . By unitarity, |||| = = , so that = for all in = = . Now let be the matrix . Then, as is easily verified, the double coset is dense in ; this is a special case of the Bruhat decomposition. Since is fixed by , the matrix coefficient is constant on . By density, = for all in . The same argument as above shows that = for all in .
(2) Suppose that is fixed by . For the unitary 1-parameter group ≅ , let be the spectral subspace corresponding to the interval . Let be the diagonal matrix with entries and for || . Then . As || tends to infinity the latter projections tend to 0 in the strong operator topology if or . Since = , it follows = in either case. By the spectral theorem, it follows that is in the spectral subspace ; in other words is fixed by . But then, by the first result, must be fixed by .
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 and its subgroups. Let be a cocompact subgroup of = } for which all non-scalar elements are hyperbolic. Let = where is the subgroup of rotations . The unit tangent bundle is = , with the geodesic flow given by the right action of and the horocycle flow by the right action of . This action if ergodic if = , i.e. the functions fixed by are just the constant functions. Since is compact, this will be the case if = . Let = . Thus acts unitarily on on the right. Any non-zero in fixed by must be fixed by , by the second result above. But in this case, if is a continuous function on of compact support with = , then = . The right hand side equals , a continuous function on . Since is right-invariant under , it follows that is constant, as required. Hence the geodesic flow is ergodic. Replacing by 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 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.
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 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 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 . The induced flow σt is given by .
The special flow corresponding to the ceiling function with base transformation is defined on the algebra B(H) given by the elements in commuting with . The induced flow corresponds to the ceiling function h ≡ 1, the constant function. Again W1, and hence commutes with I ⊗ λ(t). The special flow on B(H) is again given by . 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 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.
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 . Let = and let be the ergodic flow. Since the flow is conservative, for any projection p ≠ 0, 1 in A there is a T > 0 without σT(p) ≤ p, so that . On the other hand, as r > 0 decreases to zero
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(ar) tends to ν(p). This follows because f(t) = ν(σt(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 . Now for fixed r > 0, following , set
Set r = N–1 for N large and fN = ar. Thus 0 ≤ fN ≤ 1 in L∞(X,μ) and fN tends to a characteristic function p in L1(X,μ). But then, if ε = 1/4, it follows that χ[0,ε](fN) tends to χ[0,ε](p) = 1 – p in L1(X).Applying the same argument to 1 − fN and 1 − p, shows that if gN tends to 1 − p in L1(X) with 0 ≤ gN ≤ 1, then χ[1–ε,1](gN) tends to p in L1(X). Using the splitting A = pA ⊕ (1 − p)A, this reduces to proving that if 0 ≤ hN ≤ 1 in L∞(Y,ν) and hN tends to 0 in L1(Y,ν), then χ[1−ε,1](hN) tends to 0 in L1(Y,ν). But this follows easily by Chebyshev's inequality: indeed , so that , which tends to 0 by assumption.
Thus by definition q0(r) ∧ q1(r) = 0. Moreover, for r = N−1 sufficiently small, q0(r) ∧ σT(q1(r)) > 0. The above reasoning shows that q0(r) and q1(r) tend to 1 − p and p as r = N−1 tends to 0. This implies that q0(r)σT(q1(r)) tends to (1 − p)σT(p) ≠ 0, so is non-zero for N sufficiently large. Fixing one such N and, with r = N−1, setting q0= q0(r) and q1= q1(r), it can therefore be assumed that
The definition of q0 and q1 also implies that if δ < r/4 = (4N)−1, then
In fact if s < t
Take s = 0, so that t > 0 and suppose that e = σt(q0) ∧ q1 > 0. So e = σt(f) with f ≤ q0. Then σt(ar)e = σt(arf) ≤ 1/4 e and are ≥ 3/4 e, so that
Hence ||ar − σt(ar)||∞ ≥ 1/2. On the other hand ||ar − σt(ar)||∞ is bounded above by 2t/r, so that t ≥ r/4. Hence σt(q0) ∧ q1 = 0 if |t| ≤ δ.