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(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.

The method using representation theory relies on the following two results:^[1]

• If G = SL(2,R) acts unitarily on a Hilbert space H and ξ is a unit vector fixed by the subgroup N of upper unitriangular matrices, then ξ is fixed by G.
• If G = SL(2,R) acts unitarily on a Hilbert space H and ξ is a unit vector fixed by the subgroup A of diagonal matrices of determinant 1, then ξ is fixed by G.

(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) = (xξ,ξ) satisfies ψ(g) = 1 for g in A · N. By unitarity, ||gξ − ξ||² = 2 − ψ(g) − ψ(g^–1) = 0, so that gξ = ξ for all g in B = A · N = N · A. Now let s be the matrix ${\displaystyle {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}}$. 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 gξ = 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[s²a, s²a]. 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 ${\displaystyle {\begin{pmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{pmatrix}}}$. 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 if L²(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) 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.

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). 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 = L₁ 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.

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^∗ = 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 V₁ = 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 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 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) W₁. The induced flow corresponds to the ceiling function h ≡ 1, the constant function. Again W₁, and hence (T ⊗ I) W₁, 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.

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) ${\displaystyle \uparrow }$ 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) ⊗ ${\displaystyle \ell ^{2}}$ 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(${\displaystyle \ell ^{2}}$). 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(${\displaystyle \ell ^{2}}$) 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) ∨ σ_2t (p) ∨ σ_3t (p) ∨ ⋅⋅⋅ In particular ∨_±t>0 σ_t (p) = 1 for p ≠ 0.

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 Ambrose (1941) 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

${\displaystyle a_{r}={1 \over r}\int _{0}^{r}\sigma _{t}(p)\,dt\rightarrow p}$

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.^[2]

Note that 0 ≤ a_r ≤ 1. Now for fixed r > 0, following Ambrose (1941), set

${\displaystyle q_{0}(r)=\chi _{[0,1/4]}(a_{r}),\,\,\,q_{1}(r)=\chi _{[3/4,1]}(a_{r}).}$

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).^[3] 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)) ≤ (1−ε)⁻¹ ν(h_N), which tends to 0 by assumption.

Thus by definition q₀(r) ∧ q₁(r) = 0. Moreover, for r = N⁻¹ sufficiently small, q₀(r) ∧ σ_T(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)σ_T(q₁(r)) tends to (1 − p)σ_T(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

${\displaystyle q_{0}\wedge q_{1}=0,\,\,\,\,q_{0}\wedge \sigma _{T}(q_{1})>0.}$

The definition of q₀ and q₁ also implies that if δ < r/4 = (4N)⁻¹, then

${\displaystyle \sigma _{t}(q_{0})\wedge q_{1}=0\,\,\,\mathrm {for} \,\,\,|t|\leq \delta .}$

In fact if s < t

${\displaystyle \|\sigma _{t}(a_{r})-\sigma _{s}(a_{r})\|_{\infty }=r^{-1}\left\|\int _{r+s}^{r+t}\sigma _{x}(p)\,dx-\int _{s}^{t}\sigma _{x}(p)\,dx\right\|_{\infty }\leq {2|t-s| \over r}.}$

Take s = 0, so that t > 0 and suppose that e = σ_t(q₀) ∧ q₁ > 0. So e = σ_t(f) with f ≤ q₀. Then σ_t(a_r)e = σ_t(a_rf) ≤ 1/4 e and a_re ≥ 3/4 e, so that

${\displaystyle a_{r}-\sigma _{t}(a_{r})\geq a_{r}e-\sigma _{t}(a_{r})e\geq {3 \over 4}e-{1 \over 4}e={1 \over 2}e.}$

Hence ||a_r − σ_t(a_r)||_∞ ≥ 1/2. On the other hand ||a_r − σ_t(a_r)||_∞ is bounded above by 2t/r, so that t ≥ r/4. Hence σ_t(q₀) ∧ q₁ = 0 if |t| ≤ δ.

The elements a_r depend continuously in operator norm on r on (0,1]; from the above σ_t(a_r) is norm continuous in t. Let B₀ the closure in the operator norm of the unital *-algebra generated by the σ_t(a_r)'s. 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. By definition B₀ is a subalgebra of A and its closure B in the weak or strong operator topology can be identified with L^∞(Z,μ) where μ is also used for the restriction of μ to B. The subalgebra B is invariant under the flow σ_t, 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 (so that A will temporarily be assumed to coincide with B) and then later extended to A.^[4]

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 σ_t as t runs over the positive or negative reals is conull (i.e. the complement has measure zero). Replacing X by their intersection, an open set, it can be assumed that these unions exhaust the whole space (which will now be locally compact instead of compact). Since the flow is recurrent any orbit of σ_t 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(x) = 1/2, f(σ_t(x)) > 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 indefinitely large negative and positive times. Thus each orbit is broken up into countably many half-open intervals [r_n(x),r_n+1(x)) of length at least δ/4 with r_n(x) tending to ±∞ as n tends to ±∞. This partitioning can be normalised so that r₀(x) ≤ 0 and r₁(x) > 0. In particular if x lies in Ω, then t₀ = 0. The function r_n(x) is called the nth return time to Ω.

The cross-section Ω is a Borel set because on each compact set {σ_t(x)} with t in [N⁻¹,δ/4] with N > 4/δ, the function g(t) = f(σ_t(x)) 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 r_n are Borel functions on X. Given y in Ω, the invertible Borel transformation T is defined on Ω by S(y) = σ_t(y) where t = r₁(y), the first return time to Ω. The functions r_n(y) restrict to Borel functions on Ω and satisfy the cocycle relation:

${\displaystyle r_{m+n}=r_{m}+\tau ^{m}(r_{n}),}$

where τ is the automorphism induced by T. The hitting number N_t(x) for the flow S_t on X is defined as the integer N such that t lies in [r_N(x),r_N+1(x)). It is an integer-valued Borel function on R × X satisfying the cocycle identity

${\displaystyle N_{s+t}=N_{s}+\sigma _{s}(N_{t}).}$

The function h = r₁ is a strictly positive Borel function on Ω so formally the flow can be reconstructed from the transformation T using h a ceiling function. The missing T-invariant measure class on Ω will be recovered using the second cocycle N_t. Indeed, the discrete measure on Z defines a measure class on the product Z × X and the flow S_t on the second factor extends to a flow on the product given by

${\displaystyle \rho _{t}(m,x)=(m+N_{t}(x),S_{t}(x)).}$

Likewise the base transformation T induces a transformation R on R × Ω defined by

${\displaystyle R(s,y)=(s-h(y),T(y)).}$

These transformations are related by an invertible Borel isomorphism Φ from R × Ω onto Z × X defined by

${\displaystyle \Phi (t,y)=(N_{t}(y),S_{t}(y)).}$

Its inverse Ψ from Z × X onto R × Ω is defined by

${\displaystyle \Psi (m,x)=(-r_{-m}(y),S_{r_{-m}(y)}y).}$

Under these maps the flow R_t 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. It suffices to check that the measure class on Z × X carries over onto the same measure class as some produce measure m × ν on R × Ω, where m is Lebesgue measure and ν is a probability measure on Ω with measure class invariant under T. The measure class on Z × X is invariant under R, so defines a measure class on R × Ω, invariant under translation on the first factor. On the other hand, the only measure class on R invariant under translation is Lebesgue measure, so the measure class on R × Ω is equivalent to that of m × ν for some probability measure on Ω. By construction ν is quasi-invariant under T. Unravelling this construction, it follows that the original flow is isomorphic to the flow built under the ceiling function h for the base transformation T on (Ω,ν).^[5][6][7]

The above reasoning was made with the assumption that B = A. In general A is replaced by a norm closed separable unital *-subalgebra A₀ containing B₀, invariant under σ_t and such that σ_t(f) is a norm continuous function of t for any f in A₀. To construct A₀, first take a generating set for the von Neumann algebra A formed of countably many projections invariant under σ_t with t rational. Replace each of this countable set of projections by averages over intervals [0,N⁻¹] with respect to σ_t. The norm closed unital *-algebra that these generate yields A₀. By definition it contains B₀ = C(Y). By the Gelfand-Naimark theorem A₀ has the form C(X). The construction with a_r 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 a_r 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 (Y,λ) and an ergodic action of Z × R on M = L^∞(Y,λ) given by commuting actions τⁿ and σ_t such that there is a τ-invariant subalgebra of M isomorphic to ${\displaystyle \ell ^{\infty }}$(Z) and a σ-invariant subalgebra of M isomorphic to L^∞(R). The original ergodic flow is given by the restriction of σ to M^τ and the corresponding base transformation given by the restriction of τ to M^σ.^[8][9]

Given a flow, it is possible to describe how two different single base transformations that can be used to construct the flow are related.^[10] be transformed back into an action of Z on Y, i.e. into an invertible transformation T_Y on Y. Set-theoretically T_Y (x) is defined to be T^m(x) where m ≥ 1 is the smallest integer such that T^m(x) lies in X. It is straightforward to see that applying the same process to the inverse of T yields the inverse of T_Y. The construction can be described measure theoretically as follows. Let e = χ_Y in B = L^∞(X,ν) with ν(e) ≠ 0. Then e is an orthogonal sum of projections e_n defined as follows:

${\displaystyle e_{1}=e\tau ^{-1}(e),\,\,e_{2}=e(1-\tau ^{-1}(e))\tau ^{-2}(e),\,\,e_{3}=e(1-\tau ^{-1}(e))(1-\tau ^{-2}(e))\tau ^{-3}(e),...}$

Then if f lies in e_n B, the corresponding automorphism is τ_e(f) = τⁿ(f).

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.

• Anosov flow
• Axiom A

• von Neumann, John (1932), "Zur Operatorenmethode In Der Klassischen Mechanik", Annals of Mathematics (in German), 33 (3): 587–642, doi:10.2307/1968537, JSTOR 1968537
• Morse, Marston (1966), Lectures on Symbolic Dynamics, 1937–1938, Mimeographed Notes by Rufus Oldenburger, Institute for Advanced Study
• Hopf, Eberhard (1939), "Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung", Leipzig Ber. Verhandl. Sächs. Akad. Wiss., 91: 261–304
• Ambrose, Warren (1941), "Representation of ergodic flows", Ann. of Math., 42 (3): 723–739, doi:10.2307/1969259, JSTOR 1969259
• Ambrose, Warren; Kakutani, Shizuo (1942), "Structure and continuity of measurable flows", Duke Math. J., 9: 25–42, doi:10.1215/s0012-7094-42-00904-9
• Rohlin, V. A. (1966), "Selected topics from the metric theory of dynamical systems", Ten papers on functional analysis and measure theory, American Mathematical Society Translations. Series 2, vol. 49, American Mathematical Society, pp. 171–240
• Fomin, Sergei V.; Gelfand, I. M. (1952), "Geodesic flows on manifolds of constant negative curvature", Uspekhi Mat. Nauk, 7 (1): 118–137
• Mautner, F. I. (1957), "Geodesic flows on symmetric Riemann spaces", Ann. Math., 65 (3): 416–431, doi:10.2307/1970054, JSTOR 1970054
• Riesz, Frigyes; Sz.-Nagy, Béla (1955), Functional analysis, translated by Leo F. Boron, Frederick Ungar
• Moore, C. C. (1966), "Ergodicity of flows on homogeneous spaces", Amer. J. Math., 88 (1): 154–178, doi:10.2307/2373052, JSTOR 2373052
• Mackey, George W. (1966), "Ergodic theory and virtual groups", Math. Ann., 166 (3): 187–207, doi:10.1007/BF01361167, S2CID 119738592
• Mackey, George W. (1978), "Ergodic theory", Unitary group representations in physics, probability, and number theory, Mathematics Lecture Note Series, vol. 55, Benjamin/Cummings Publishing Co, pp. 133–142, ISBN 0805367020
• Mackey, George W. (1990), "Von Neumann and the Early Days of Ergodic Theory", in Glimm, J.; Impagliazzo, J.; Singer, I. (eds.), The Legacy of John von Neumann, Proceedings of Symposia in Pure Mathematics, vol. 50, American Mathematical Society, pp. 34~47, ISBN 9780821814871
• Krengel, Ulrich (1968), "Darstellungssätze für Strömungen und Halbströmungen I", Math. Annalen (in German), 176 (3): 181–190, doi:10.1007/bf02052824, S2CID 124603266
• Kubo, Izumi (1969), "Quasi-flows", Nagoya Math. J., 35: 1–30, doi:10.1017/s002776300001299x
• Howe, Roger E.; Moore, Calvin C. (1979), "Asymptotic properties of unitary representations", J. Funct. Anal., 32: 72–96, doi:10.1016/0022-1236(79)90078-8
• Cornfeld, I. P.; Fomin, S. V.; Sinaĭ, Ya. G. (1982), Ergodic theory, Grundlehren der Mathematischen Wissenschaften, vol. 245, translated by A. B. Sosinskiĭ, Springer-Verlag, ISBN 0-387-90580-4
• Zimmer, Robert J. (1984), Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser, ISBN 3-7643-3184-4
• Bedford, Tim; Keane, Michael; Series, Caroline, eds. (1991), Ergodic theory, symbolic dynamics and hyperbolic spaces, Oxford University Press, ISBN 019853390X
• Adams, Scot (2008), "Decay to zero of matrix coefficients at adjoint infinity", Group representations, ergodic theory, and mathematical physics: a tribute to George W. Mackey, Contemp. Math., vol. 449, Amer. Math. Soc., pp. 43–50
• Moore, C. C. (2008), "Virtual groups 45 years later", Group representations, ergodic theory, and mathematical physics: a tribute to George W. Mackey, Contemp. Math., vol. 449, Amer. Math. Soc., pp. 267~300
• Pedersen, Gert K. (1979), C^∗-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, ISBN 0-12-549450-5
• Varadarajan, V. S. (1985), Geometry of quantum theory (Second ed.), Springer-Verlag, ISBN 0-387-96124-0
• Takesaki, M. (2003), Theory of operator algebras, II, Encyclopaedia of Mathematical Sciences, vol. 125, Springer-Verlag, ISBN 3-540-42914-X
• Takesaki, M. (2003a), Theory of operator algebras, III, Encyclopaedia of Mathematical Sciences, vol. 127, Springer-Verlag, ISBN 3-540-42913-1
• Morris, Dave Witte (2005), Ratner's theorems on unipotent flows, Chicago Lectures in Mathematics, University of Chicago Press, arXiv:math/0310402, Bibcode:2003math.....10402W, ISBN 0-226-53983-0
• Nadkarni, M. G. (2013), Basic ergodic theory, Texts and Readings in Mathematics, vol. 6 (Third ed.), Hindustan Book Agency, ISBN 978-93-80250-43-4