Ergodic flow


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, 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.

Ergodicity of geodesic and horocycle flows

The method using representation theory relies on the following two results:
To prove the first result, let S be the matrix. Thus is the subgroup of lower unitriangular matrices. In this case the product is is dense in G. Let η = Sξ. Then η is fixed by. Hence for g in , the matrix coefficient defined by φ = satisfies φ =. By density φ = φ = 1 for g in G. In particular if ψ = satisfies ψ = 1 for all g. But then ||gξ − ξ||2 = 2 − ψ − ψ = 0, so that gξ = ξ for all g in G.
To prove the second result, suppose that ξ is fixed by A. For the unitary 1-parameter group N ≅ R, let P be the spectral subspace corresponding to the interval . Let g be the diagonal matrix with entries s and s−1 for s > 1. Then. 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 gξ = ξ, it follows Pξ = 0 in either case. By the spectral theorem, it follows that ξ is in the spectral subspace P; in other words ξ is fixed by N. But then, by the first result, ξ must be fixed by G.
Let Γ be a cocompact subgroup of PSL = G / for which all non-scalar elements are hyperbolic. Let X = Γ \ G / K where K is the subgroup of rotations. 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 left action of N. This action if ergodic if
L∞A = C, i.e. the functions fixed by A are just the constant functions. Since Γ \ G is compact, this will be the case if L2A = C. Let H = L2. 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 gξ 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 giving rise to an automorphism τ of A = L∞. This gives rise to an invertible transformation T ⊗ id of the measure space, where m is Lebesgue measure, and hence an automorphism τ ⊗ id of A L∞. Translation Lt defines a flow on R preserving m and hence a flow λt on L∞. Let S = L1 with corresponding automorphism σ of L∞. Thus τ ⊗ σ gives an automorphism of A L∞ which commutes with the flow id ⊗ λt. The induced measure space Y is defined by B = L∞ = L∞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∞, it follows that the functions fixed by the flow can be identified with L∞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, a unitary operator such that RλR∗ = Vt where λ is translation by t and Vt is multiplication by eitx. Thus Vt lies in L∞. 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∞, acting on L2 ⊗ L2. In particular W commutes with I ⊗Vt. So commutes with I ⊗ λ. The action T on L∞ induces a unitary U on L2 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 h with base transformation T is defined on the algebra B 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. The special flow on B 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.
If St is an ergodic flow on the measure space corresponding to a 1-parameter group of automorphisms σt of A = L∞, then by the Hopf decomposition either every St with t ≠ 0 is dissipative or every St with t ≠ 0 is conservative. In the former case the ergodic flow is transitive, so that A = L∞ with Lebesgue measure and R acting by translation.
To handle the dissipative case, note that A = L∞ is a maximal Abelian von Neumann algebra on L2. 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 for t > 0 and λ = t. In this case σt =E) where E is a projection-valued measure on R. These projections generate a von Neumann subalgebra B of A. By ergodicity σt 1 as t tends to −∞. The Hilbert space L2 can be identified with the completion of the subspace of f in A with λ < ∞. The subspace corresponding to B can be identified with L2 and B with L∞. 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 admits a decomposition L2 ⊗ 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 in B, the entries of a can be taken to be in L∞ ∩ L1. For such functions f, as an elementary case of the ergodic theorem the average of σt over tends in the weak operator topology to ∫ f dt. Hence for appropriate χ 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∞.
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 ∨ σ2t ∨ σ3t ∨ ⋅⋅⋅ In particular ∨±t>0 σt = 1 for p ≠ 0.
Theorem. 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.
In the dissipative case, A = L∞ with R acting by translation, which is just the induced action from the ergodic translation action of Z on, corresponding to the atomic measure space Z with the discrete measure. So it can be assumed that the flow is
properly ergodic. In particular for any p ≠ 0, 1 in A there is a T > 0 without σT ≤ p, so that. On the other hand as r > 0 decreases to zero
Indeed it suffices to show that if ν is any finite measure on A, then ν tends to ν. This follows because f = ν is a continuous function of t so that the average of f over tends to f as r tends to 0.
Note that 0 ≤ ar ≤ 0. Now for fixed r > 0, set
Thus by definition q0 ∧ q1 = 0. Moreover for r = N−1 sufficiently small, q0 ∧ σT > 0.
To see this it suffices to show that if 0 ≤ fN ≤ 1 in L∞, 0 < ε < 1 and fN tends to a characteristic function Q in L1, then χ tends to χ = 1 – Q in L1. Using the splitting A = QA ⊕ A, this reduces to proving that if 0 ≤ hN ≤ 1 in L∞ and hN tends to 0 in L1, then χ tends to 0 in L1. But this follows easily by Chebyshev's inequality: indeed, so that, which tends to 0 by assumption.
The above argument shows that q0 and q1 tend to 1 − p and p as r = N−1 tends to 0. This implies that q0σT tends to σT ≠ 0, so is non-zero for N sufficiently large. Fixing one such N and, with r = N−1, setting q0= q0 and q1= q1, it can therefore be assumed that
The definition of q0 and q1 also implies that if δ < r/4 = −1, then
In fact if s < t
Take s = 0, so that t > 0 and suppose that e = σt ∧ q1 > 0. So e = σt with f ≤ q0. Then σte = σt ≤ 1/4 e and are ≥ 3/4 e, so that
Hence ||ar − σt||∞ ≥ 1/2. On the other hand ||ar − σt||∞ is bounded above by 2t/r, so that t ≥ r/4. Hence σt ∧ q1 = 0 if |t| ≤ δ.
The elements ar depend continuously in operator norm on r on is norm continuous in t. Let B0 the closure in the operator norm of the unital *-algebra generated by the σt's. It is commutative and separable so, by the Gelfand–Naimark theorem, can be identified with C where Z is its spectrum, a compact metric space. By definition B0 is a subalgebra of A and its closure B in the weak or strong operator topology can be identified with L∞ where μ is also used for the restriction of μ to B. The subalgebra B is invariant under the flow σt, which is therefore ergodic. The analysis of this action on B0 and B yields all the tools necessary for constructing the ergodic transformation T and ceiling function h. This will first be carried out for B and then later extended to A.
The projections q0 and q1 correspond to characteristic functions of open sets. X0 and X1 The assumption of proper ergodicity implies that the union of either of these open sets under translates by σt as t runs over the positive or negative reals is conull. Replacing X by their intersection, an open set, it can be assumed that these unions exhaust the whole space. Since the flow is recurrent any orbit of σt passes through both sets infinitely many times as t tends to either +∞ or −∞. Between a spell first in X0 and then in X1 f must assume the value 1/2 and then 3/4. The last time f equals 1/2 to the first time it equals 3/4 must involve a change in t of at least δ/4 by the Lipschitz continuity condition. Hence each orbit must intersect the set Ω of x for which f = 1/2, f > 1/2 for 0 < t ≤ δ/4 infinitely often. The definition implies that different insections with an orbit are separated by a distance of at least δ/4, so Ω intersects each orbit only countably many times and the intersections occur at indefinielty large negative and positive times. Thus each orbit is broken up into countably many half-open intervals with N > 4/δ, the function g = f has an infimum greater than 1/2 + M−1 for a sufficiently large integer M. Hence Ω can be written as a countable intersection of sets, each of which is a countable unions of closed sets; so Ω is therefore a Borel set. This implies in particular that the functions rn are Borel functions on X. Given y in Ω, the invertible Borel transformation T is defined on Ω by S = σt where t = r1, the first return time to Ω. The functions rn restrict to Borel functions on Ω and satisfy the cocycle relation:
where τ is the automorphism induced by T. The hitting number Nt for the flow St on X is defined as the integer N such that t lies in with respect to
σt. The norm closed unital *-algebra that these generate yields A0. By definition it contains B0 = C. By the Gelfand-Naimark theorem A0 has the form C. The construction with ar above applies equally well here: indeed since B0 is a subalgebra of A0, Y is a continuous quotient of X, so a function such as ar is equally well a function on X. The construction therefore carries over mutatis mutandis to A, through the quotient map.
In summary there exists a measure space and an ergodic action of Z × R on M = L∞ given by commuting actions τn and σt such that there is a τ-invariant subalgebra of M isomorphic to and a σ-invariant subalgebra of M isomorphic to L∞. The original ergodic flow is given by the restriction of σ to Mτ and the corresponding base transformation given by the restriction of τ to Mσ.
Given a flow, it is possible to describe how two different single base transformations that can be used to construct the flow are related. be transformed back into an action of Z on Y, i.e. into an invertible transformation TY on Y. Set-theoretically TY is defined to be Tm where m ≥ 1 is the smallest integer such that Tm lies in X. It is straightforward to see that applying the same process to the inverse of T yields the inverse of TY. The construction can be described measure theoretically as follows. Let e = χY in B = L∞ with ν ≠ 0. Then e is an orthogonal sum of projections en defined as follows:
Then if f lies in en B, the corresponding automorphism is τe = τn.
With these definitions two ergodic transformations τ1, τ2 of B1 and B2 arise from the same flow provided there are non-zero projections e1 and e2 in B1 and B2 such that the systems e1, e1B1 and e2, e2B2 are isomorphic.