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Appendix. Ergodicity of the Geodesic Flow

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Appendix. Ergodicity of the Geodesic Flow Appendix. Ergodicity of the geodesic flow M. BRIN We present here a reasonably self-contained and short proof of the ergodicity of the geodesic flow on a compact n-dimensional manifold of strictly negative sectional curvature. The only complete proof of specifically this fact was published by D.Anosov in [Anol]. The ergodicity of the geodesic flow is buried there among quite a number of general properties of hyperbolic systems, heavily depends on most of the rest of the results and is rather difficult to understand, especially for somebody whose background in smooth ergodic theory is not very strong. Other works on this issue either give only an outline of the argument or deal with a considerably more general situation and are much more difficult. 1. Introductory remarks E.Hopf (see [HoI], [Ho2]) proved the ergodicity of the geodesic flow for com­ pact surfaces of variable negative curvature and for compact manifolds of constant negative sectional curvature in any dimension. The general case was established by D.Anosov and Ya.Sinai (see [Anol], [AnSi]) much later. Hopf's argument is rel­ atively short and very geometrical. It is based on the property of the geodesic flow which Morse called "instability" and which is now commonly known as "hyper­ bolicity". Although the later proofs by Anosov and Sinai follow the main lines of Hopf's argument, they are considerably longer and more technical. The reason for the over 30 year gap between the special and general cases is, in short, that length is volume in dimension I but not in higher dimensions. More specifically, these ergodicity proofs extensively use the horosphere H(v) as a function of the tangent vector v and the absolute continuity of a certain map p between two horospheres (the absolute continuity of the horospheric foliations). Since the horosphere H(v) depends on the infinite future of the geodesic "tv determined by v, the function H (.) is continuous but, in the general case, not differentiable even if the Riemannian metric is analytic. As a result, the map p is, in general, only continuous. For n = 2 the horospheres are I-dimensional and the property of bounded volume distortion 82 Appendix: Ergodicity of the geodesic flow for p is equivalent to its Lipschitz continuity which holds true in special cases but is false in general. The general scheme of the ergodicity argument below is the same as Hopf's. Most ideas in the proof of the absolute continuity of the stable and unstable foliations are the same as in [Anol] and [AnSi]. However, the quarter century de­ velopment of hyperbolic dynamical systems and a couple of original elements have made the argument less painful and much easier to understand. The proof goes through with essentially no changes for the case of a manifold of negative sectional curvature with finite volume, bounded and bounded away from 0 curvature and bounded first derivatives of the curvature. For surfaces this is exactly Hopf's result. The assumption of bounded first derivatives cannot be avoided in this argument. The proof also works with no changes for any manifold whose geodesic flow is Anosov. In addition to being ergodic, the geodesic flow on a compact manifold of negative sectional curvature is mixing of all orders, is a K-flow, is a Bernoulli flow and has a countable Lebesgue spectrum, but we do not address these stronger properties here. In Section 2 we list several basic facts from measure and ergodic theory. Continuous foliations with C1-leaves and their absolute continuity are discussed in Section 3. In Section 4 we recall the definition of an Anosov flow and some of its basic properties and prove the Holder continuity of the stable and unstable distributions. The absolute continuity of the stable and unstable foliations for the geodesic flow and its ergodicity are proved in Section 5. 2. Measure and ergodic theory preliminaries A measure space is a triple (X, m, p,), where X is a set, m is a O"-algebra of measurable sets and /-l is a O"-additive measure. We will always assume that the measure is finite or O"-finite. Let (X, m, p,) and (Y, 123, v) be measure spaces. A map 7/J : X ~ Y is called measurable if the preimage of any measurable set is measurable; it is called nonsingular if, in addition, the preimage of a set of measure 0 has measure O. A nonsingular map from a measure space into itself is called a nonsingular transformation (or simply a transformation). Denote by A the Lebesgue measure on JR. A measurable flow ¢ in a measure space (X, m, p,) is a map ¢ : X x JR ~ X such that (i) ¢ is measurable with respect to the product measure fJ x A on X x JR and measure /-l on X and (ii) the maps ¢t(.) = ¢(', t) : X ~ X form a one-parameter group of transformations of X, that is ¢t 0 ¢s = ¢t+s for any t, s E R The general discussion below is applicable to the discrete case t E Z and to a measurable action of any locally compact group. A measurable function f : X ~ JR is ¢-invariant if /-l({X EX: f(¢tx) #­ f(x)}) = 0 for every t ERA measurable set B E m is ¢-inva'riant if its charac­ teristic function IB is ¢-invariant. A measurable function f : X ~ JR is strictly ¢-invariant if f (¢t x) = f (x) for all x EX, t ERA measurable set B is strictly ¢-invariant if IB is strictly ¢-invariant. Measure and ergodic theory preliminaries 83 We say that something holds true mod 0 in X or for /-l- a. e. x if it holds on a subset of full /-l-measure in X. A null set is a set of measure O. 2.1 Proposition. Let ¢> be a measurable flow in a measure space (X,~, /-l) and let f : X -+ lR be a ¢>-invariant function. Then there is a strictly ¢>-invariant measurable function j such that f(x) = j(x) mod o. Proof. Consider the measurable map cI> : X x lR -+ X, cI>(x, t) = ¢>tx, and the product measure 1/ in X x R Since f is ¢>-invariant, 1/( {(x, t) : f(¢>tx) = f(x)}) = 1. By the Fubini theorem, the set Af = {x EX: f(¢>tx) = f(x) for a.e. t E lR} has full /-l-measure. Set j(x) = {fo(Y) if ¢>tx = y E Af for some t E lR otherwise. If ¢>tx = Y E Af and ¢>BX = Z E Af then clearly f(y) = f(z). Therefore j is well defined and strictly ¢>-invariant. 0 2.2 Definition (Ergodicity). A measurable flow ¢> in a measure space (X,~, /-l) is called ergodic if any ¢>-invariant measurable function is constant mod 0, or equivalently, if any measurable ¢>-invariant set has either full or 0 measure. The equivalence of the two definitions follows directly from the density of the linear combinations of step functions in bounded measurable functions. A measurable flow ¢> is measure preserving (or /-l is ¢>-invariant) if /-l(¢>t(B)) = /-l(B) for every t E lR and every B E~. 2.3 Ergodic Theorems. Let ¢> be a measure preserving flow in a finite measure space (X,~, /-l). For a measurable function f : X -+ lR set 1 T 1 T fi(x) = T ior f(¢>t x ) dt and fT(x) = T ior f(¢>-t x ) dt. (1) (Birkhoff) If f E Ll(X,~, /-l) then the limits f+(x) = limT-->oo fi(x) and f-(x) = limT-->oo fT(x) exist and are equal for /-l-a.e. x E X, more­ over f+, f- are /-l-integrable and ¢>-invariant. (2) (von Neumann) If f E L2(X,~, /-l) then fi and fT converge in L2(X,~, /-l) to ¢>-invariant functions f+ and f-, respectively. o 2.4 Remark. If f E L2(X,~, /-l), that is, if f2 is integrable, then f+ and f- repre­ sent the same element j of L2(X,~, /-l) and 1 is the projection of f onto the sub­ space of ¢>-invariant L2-functions. To see this let h be any ¢>-invariant L2-function. 84 Appendix: Ergodicity of the geodesic flow Since cP preserves /l we have (f(x) -J(x))hd/l(x) f(x)h(x)d/l(x) lim ~ T f(cPtx)h(x)dtd/l(x) 1x =1x -1xT->ooT Jro T =1 f(x)h(x)d/l(x) - r lim -T1 r f(y)h(cP-1y)dtd/l(Y) X JxT->oo Jo = Lf(X)h(X)d/l(X)- Lf(Y)h(Y)d/l(Y) =0. In the topological setting invariant functions are mod 0 constant on sets of orbits converging forward or backward in time. Let (X,~, /l) be a finite measure space such that X is a compact metric space with distance d, /l is a measure positive on open sets and ~ is the /l-completion of the Borel a-algebra. Let cP be a continuous flow in X, that is the map (x, t) ---> cPtx is continuous. For x E X define the stable VS(x) and unstable VU(x) sets by the formulas VS(X)={YEX: d(cPtx,cPty)--->O as t---> oo}, (2.5) VU(x) = {y EX: d(cPtx, cPty) ---> 0 as t ---> -oo}. 2.6 Proposition. Let cP be a continuous flow in a compact metric space X preserving a finite measure /l positive on open sets and let f : X ---> IR be a cP-invariant measurable function. Then f is mod 0 constant on stable sets and unstable sets, that is, there are null sets Ns and Nu such that fey) = f(x) for any X,y E X \ Ns, y E VS(x) and fez) = f(x) for any x, z E X \ Nu , z E VU(x).
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