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.
Then, as is easily verified, the double coset BsB is dense in G; this is a special case of the Bruhat decomposition.
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.
But in this case, if f is a continuous function on G of compact support with ∫ f = 1, then ξ = ∫ f(g) gξ dg.
Replacing A by N and using the first result above, the same argument shows that the horocycle flow is ergodic.
Examples of flows induced from non-singular invertible transformations of measure spaces were defined by von Neumann (1932) 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).
Thus τ ⊗ σ gives an automorphism of A ⊗ L∞(R) which commutes with the flow id ⊗ λt.
Since λt acts ergodically on L∞(R), it follows that the functions fixed by the flow can be identified with L∞(X)τ.
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) W1.
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.
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.
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) ⊗
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.
But such an element commutes with Ut so is fixed by σt, contradicting ergodicity.
If the flow leaves a probability measure invariant, the same is true of the base transformation.
For simplicity only the original result of Ambrose (1941) is considered, the case of an ergodic flow preserving a probability measure μ.
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.
It is commutative and separable so, by the Gelfand–Naimark theorem, can be identified with C(Z) where Z is its spectrum, a compact metric space.
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 (so that A will temporarily be assumed to coincide with B) and then later extended to A.
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 (i.e. the complement has measure zero).
Since the flow is recurrent any orbit of σt passes through both sets infinitely many times as t tends to either +∞ or −∞.
The function rn(x) is called the nth return time to Ω.
The functions rn(y) restrict to Borel functions on Ω and satisfy the cocycle relation: where τ is the automorphism induced by T. The hitting number Nt(x) for the flow St on X is defined as the integer N such that t lies in [rN(x),rN+1(x)).
The missing T-invariant measure class on Ω will be recovered using the second cocycle Nt.
Indeed, the discrete measure on Z defines a measure class on the product Z × X and the flow St on the second factor extends to a flow on the product given by Likewise the base transformation T induces a transformation R on R × Ω defined by These transformations are related by an invertible Borel isomorphism Φ from R × Ω onto Z × X defined by Its inverse Ψ from Z × X onto R × Ω is defined by Under these maps the flow Rt is carried onto translation by t on the first factor of R × Ω and, in the other direction, the invertible R is carried onto translation by -1 on Z × X.
To construct A0, first take a generating set for the von Neumann algebra A formed of countably many projections invariant under σt with t rational.
Replace each of this countable set of projections by averages over intervals [0,N−1] with respect to σt.
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 (τ1)e1, e1B1 and (τ2)e2, e2B2 are isomorphic.