Ergodic Flow
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, ergodic flows occur in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, through the
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
and
horocycle In hyperbolic geometry, a horocycle (), sometimes called an oricycle, oricircle, or limit circle, is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional case of a horospher ...
flows of closed
hyperbolic surface In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P' ...
s. Both of these examples have been understood in terms of the theory of
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
s of
locally compact group In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are lo ...
s: if Γ is the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of a
closed surface In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as ...
, regarded as a discrete subgroup of the
Möbius group Moebius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Theodor Möbius (1821–1890), German philologist * Karl Möbius (1825–1908), German zoologist and ecologist * Paul ...
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 In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
''G'' / Γ. The Ambrose-Kakutani theorem expresses every ergodic flow as the flow built from an invertible ergodic transformation on a
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
using a ceiling function. In the case of
geodesic flow In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
, the ergodic transformation can be understood in terms of
symbolic dynamics In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (e ...
; and in terms of the ergodic actions of Γ on the boundary ''S''1 = ''G'' / ''AN'' and ''G'' / ''A'' = ''S''1 × ''S''1 \ diag ''S''1. Ergodic flows also arise naturally as invariants in the classification of
von Neumann algebra In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algeb ...
s: the flow of weights for a factor of type III0 is an ergodic flow on a
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
.


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 In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
= 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 \begin 0 & 1 \\ -1 & 0 \end. Then, as is easily verified, the double coset is dense in ; this is a special case of the
Bruhat decomposition In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) ''G'' = ''BWB'' of certain algebraic groups ''G'' into cells can be regarded as a general expression of the principle ...
. 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 surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s 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 \begin \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end. 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 Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
and
ergodic theory Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
. 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'' = ''L''1 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.


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 ''R''λ(''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 ''V''1 = ''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 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 σ''t'' 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 ''W''1, 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 ''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 In mathematics, the Hopf decomposition, named after Eberhard Hopf, gives a canonical decomposition of a measure space (''X'', μ) with respect to an invertible non-singular transformation ''T'':''X''→''X'', i.e. a transformation which with its ...
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 In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algeb ...
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'') \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'') ⊗ \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(\ell^2). If can thus be realised as a matrix with entries in ''B''. Multiplying by χ 'r'',''s''/sub> 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 Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
the average of σ''t''(''f'') over ˆ’''R'',''R''tends in the weak operator topology to ∫ ''f''(''t'') ''dt''. Hence for appropriate χ 'r'',''s''/sub> this will produce an element in ''A'' which lies in C ⊗ B(\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'') ∨ σ2''t'' (''p'') ∨ σ3''t'' (''p'') ∨ â‹…â‹…â‹… In particular ∨±''t''>0 σ''t'' (''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 σ''T''(''p'') ≤ ''p'', so that . On the other hand, as ''r'' > 0 decreases to zero :a_r = \int_0^r \sigma_t(p) \, dt \rightarrow p in the
strong operator topology In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ...
or equivalently the
weak operator topology In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number \langle Tx, y\rangle is ...
(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 ,''r''tends to ''f''(0) as ''r'' tends to 0. Note that . Now for fixed ''r'' > 0, following , set :q_0(r)=\chi_(a_r), \,\,\, q_1(r)=\chi_(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 ,ε/sub>(''f''''N'') tends to χ ,ε/sub>(''p'') = 1 – ''p'' in L1(''X''). 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 −ε,1/sub>(''h''''N'') tends to 0 in L1(''Y'',ν). But this follows easily by
Chebyshev's inequality In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from th ...
: indeed , so that , which tends to 0 by assumption. Thus by definition ''q''0(''r'') ∧ ''q''1(''r'') = 0. Moreover, for ''r'' = ''N''−1 sufficiently small, ''q''0(''r'') ∧ σ''T''(''q''1(''r'')) > 0. The above reasoning shows that ''q''0(''r'') and ''q''1(''r'') tend to 1 − ''p'' and ''p'' as ''r'' = ''N''−1 tends to 0. This implies that ''q''0(''r'')σ''T''(''q''1(''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 ''q''0= ''q''0(''r'') and ''q''1= ''q''1(''r''), it can therefore be assumed that :q_0 \wedge q_1 =0,\,\,\,\, q_0 \wedge \sigma_T(q_1)>0. The definition of ''q''0 and ''q''1 also implies that if δ < ''r''/4 = (4''N'')−1, then :\sigma_t(q_0)\wedge q_1 = 0 \,\,\,\mathrm\,\,\, , t, \le \delta. In fact if ''s'' < ''t'' :\, \sigma_t(a_r) - \sigma_s(a_r)\, _\infty = r^\left\, \int_^ \sigma_x(p)\, dx - \int_s^ \sigma_x(p) \, dx\right\, _\infty \le . Take ''s'' = 0, so that ''t'' > 0 and suppose that ''e'' = σ''t''(''q''0) ∧ ''q''1 > 0. So ''e'' = σ''t''(''f'') with ''f'' ≤ ''q''0. Then σ''t''(''a''''r'')''e'' = σ''t''(''a''''r''''f'') ≤ 1/4 ''e'' and ''a''''r''''e'' ≥ 3/4 ''e'', so that :a_r -\sigma_t(a_r) \ge a_r e -\sigma_t(a_r)e \ge e - e= e. Hence , , ''a''''r'' − σ''t''(''a''''r''), , ∞ ≥ 1/2. On the other hand , , ''a''''r'' − σ''t''(''a''''r''), , ∞ is bounded above by 2''t''/''r'', so that ''t'' ≥ ''r''/4. Hence σ''t''(''q''0) ∧ ''q''1 = 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 ''B''0 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 representation, Gelfand–Naimark theorem, can be identified with ''C''(''Z'') where ''Z'' is its
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
, a compact metric space. By definition ''B''0 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 ''B''0 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''. The projections ''q''0 and ''q''1 correspond to characteristic functions of open sets. ''X''0 and ''X''1 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 ''X''0 and then in ''X''1 ''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 ''r''0(''x'') ≤ 0 and ''r''1(''x'') > 0. In particular if ''x'' lies in Ω, then ''t''0 = 0. The function ''r''''n''(''x'') is called the ''n''th return time to Ω. The cross-section Ω is a Borel set because on each compact set 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'' = ''r''1(''y''), the first return time to Ω. The functions ''r''''n''(''y'') restrict to Borel functions on Ω and satisfy the cocycle relation: :r_ = 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 :N_ = N_s + \sigma_s(N_t). The function ''h'' = ''r''1 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 :\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 :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 :\Phi(t,y)=(N_t(y),S_t(y)). Its inverse Ψ from Z × ''X'' onto R × Ω is defined by :\Psi(m,x)=(-r_(y), S_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). The above reasoning was made with the assumption that ''B'' = ''A''. In general ''A'' is replaced by a norm closed separable unital *-subalgebra ''A''0 containing ''B''0, invariant under σ''t'' and such that σ''t''(''f'') is a norm continuous function of ''t'' for any ''f'' in ''A''0. To construct ''A''0, 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 ''A''0. By definition it contains ''B''0 = C(''Y''). By the Gelfand-Naimark theorem ''A''0 has the form C(''X''). The construction with ''a''''r'' above applies equally well here: indeed since ''B''0 is a subalgebra of ''A''0, ''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 ''Mutatis mutandis'' is a Medieval Latin phrase meaning "with things changed that should be changed" or "once the necessary changes have been made". It remains unnaturalized in English and is therefore usually italicized in writing. It is used i ...
'' 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 τ''n'' and σ''t'' such that there is a τ-invariant subalgebra of ''M'' isomorphic to \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. 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 ''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: :e_1= e \tau^(e),\,\, e_2=e(1-\tau^(e))\tau^(e),\,\, e_3=e (1-\tau^(e))(1-\tau^(e))\tau^(e),... Then if ''f'' lies in ''e''''n'' ''B'', the corresponding automorphism is τ''e''(''f'') = τ''n''(''f''). With these definitions two ergodic transformations τ1, τ2 of ''B''1 and ''B''2 arise from the same flow provided there are non-zero projections ''e''1 and ''e''2 in ''B''1 and ''B''2 such that the systems (τ1)''e''1, ''e''1''B''1 and (τ2)''e''2, ''e''2''B''2 are isomorphic.


See also

*
Anosov flow In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "contr ...
* Axiom A


Notes


References

* * * * * * * * * * * * * * * * * * * * * * * * * * *{{citation, last=Nadkarni, first= M. G., title=Basic ergodic theory, edition=Third , series= Texts and Readings in Mathematics, volume=6, publisher=Hindustan Book Agency, year=2013, isbn= 978-93-80250-43-4 Ergodic theory