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«. UNIQUE ERGODICITY, STABLE ERGODICITY AND THE MAUTNER PHENOMENON FOR DIFFEOMORPHISMS CHARLES PUGH, MICHAEL SHUB, AND ALEXANDER STARKOV Introduction ...»

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UNIQUE ERGODICITY, STABLE ERGODICITY AND THE

MAUTNER PHENOMENON FOR DIFFEOMORPHISMS

CHARLES PUGH, MICHAEL SHUB, AND ALEXANDER STARKOV

Introduction

Beginning with [GPS] the first two authors have been studying stable ergodicity

of volume preserving partially hyperbolic diffeomorphisms on a compact manifold

M. The most recent survey on the subject is [PS3]. A key issue is the way in which the strong stable and strong unstable manifolds foliate M. To prove ergodicity one assumes essential accessibility, namely that every Borel set S ⊂ M which consists simultaneously of whole strong stable leaves and whole strong unstable leaves has measure zero or one. Such a set S is said to be us-saturated. As essential accessibility is a measure theory concept, it is difficult to verify and even more difficult to prove stable under perturbation. A stronger assumption is full accessibility1 in which it is required that M and the empty set are the only ussaturated sets. In many cases full accessibility is stable under perturbation, and this leads to stable ergodicity.

We have conjectured that the stably ergodic diffeomorphisms are open and dense among C 2 volume preserving partially hyperbolic diffeomorphisms. Our plan of attack was to prove that an open and dense subset of partially hyperbolic diffeomorphisms are fully accessible. As already noted, this seems far easier than the similar assertion for essential accesssibility, so we focused on the full accessibility property. The following recent developments, however, caused us to reconsider our position and to shift our attention more in the direction of essential accessibility.

(a) Among affine diffeomorphisms of finite volume, compact homogeneous spaces those which are stably ergodic among left translations are precisely those with the essential accessibility property [St3]. In other words, affine stable ergodicity is equivalent to essential accessibility. The proof relies significantly on the structural properties of Lie groups.

(b) As was shown by Federico Rodgriguez Hertz, essential accessibility without full accessibility sometimes leads to (nonlinear) stable ergodicity, [RH].

(c) The Mautner phenomenon from representation theory leads to a proof of half of the affine stable ergodicity result mentioned in (a), while in [PS2] we establish a nonlinear version of the Mautner phenomenon, which we apply to nonlinear stable ergodicity.

Below, we give a proof of the Mautner phenomenon in the case it is used for (a), but instead of structural properties of Lie groups or their representation theory, The second author was supported in part by NSF Grant #DMS-9988809.

The third author was supported by the Leading Scientific School Grant, No 457.2003.1.

1 In previous papers, we referred to full accessibility as us-accessibility, and to a stronger condition as homotopy accessibility. The latter is always stable under perturbation and is often a consequence of the former.

2 CHARLES PUGH, MICHAEL SHUB, AND ALEXANDER STARKOV

we use Birkhoff’s theorem as in the Hopf-Anosov argument for the ergodicity of Anosov systems. This proof makes us feel we have landed in the right place.

We would like to see a unified explanation of these Mautner phenomena, one that might generalize Rodriguez-Hertz’s theorem to all essentially accessible affine diffeomorphisms. To this end we wondered what more of a potentially useful nature could be said about the strong stable and unstable manifold foliations. The third author has extended the results in his monograph [St1], and answered question 6.8 of [BPSW] for affine diffeomorphisms – namely, in the affine, essentially accessible case, the strong stable or strong unstable manifold foliations are uniquely ergodic.

See Theorem 0.7 below.

The Mautner phenomenon Roughly speaking the Mautner phenomenon refers to invariance of a function along trajectories of one flow implying invariance along certain transverse flows.

Mautner first observed the phenomenon in an affine ergodicity proof – invariance of a function along the geodesic flow (for a compact surface of constant negative curvature) implies invariance along the horocycle flows. It has been generalized considerably by Auslander-Green, Dani and Moore for the ergodic theory of flows on homogeneous spaces. See [St1] for references, proofs and a discussion of the results.

A version the Mautner phenomenon applies precisely to prove the ergodicity of the essentially accessible affine diffeomorphisms of finite volume, compact homogeneous spaces. In [PS3] we sketched a proof of this result. Below, we do a better job. The proof is quite close in structure to the best proof we have for partially hyperbolic diffeomorphisms with the essential accessibility property, see [PS2] and [PS3].

Let G be a connected Lie group, and B ⊂ G a closed subgroup such that G/B is compact and of finite volume, i.e., G/B admits finite G-invariant volume.2 Let f ∈ Aff(G/B) be an affine map of G/B, i.e., f = La ◦ A, where La : G/B → G/B is left transation by a fixed element a ∈ G and A : G/B → G/B is a map induced by a fixed automorphism A ∈ Aut(G) such that A(B) = B. The covering map f = La ◦ A : G → G induces an automorphism df of the Lie algebra g. With respect to df, the Lie algebra g splits into generalized eigenspaces g = gu ⊕ gc ⊕ gs such that the eigenvalues of df are respectively outside, on, or inside the unit circle.





The corresponding connected subgroups Gu, Gc, and Gs are the unstable, neutral, and stable horospherical subgroups. Their orbits form the (strong) unstable, center, and (strong) stable foliations for f on G/B. These facts are proved in [PSS].

Let H ⊂ G be the subgroup generated by Gu and Gs. It is normal and called the hyperbolic subgroup for f. See [PS1]. Under the previous conditions, the

version of the Mautner phenomenon that we prove is:

2 “Finite volume” includes the requirement that the measure on G/B be G-invariant. In particular, R) (a) If Γ is a uniform discrete subgroup of G = SL(2, then the homogeneous space G/Γ is of finite volume, but (b) if T is the subgroup of G consisting of upper triangular matrices then the homogeneous space G/T ≈ S 1 is not of finite volume because there is no G-invariant measure on it.

UNIQUE ERGODICITY, STABLE ERGODICITY AND THE MAUTNER PHENOMENON FOR DIFFEOMORPHISMS

–  –  –

the limit exists almost everywhere, and Inv1 (f ) is the space of L1 invariant functions. Clearly β(φ) = φ. If ψ is continuous then it is straightforward to see that if β(ψ)(x) exists at one point of a strong stable manifold W ss (p) then it exists at all points of W ss (p) and has the same value. Thus, β(ψ) is essentially constant along the leaves of the strong stable foliation.

The space C 0 (M, R) is dense in L1 (M, R), and so there is a sequence of continuos functions ψn that converges to φ in the L1 sense. Since β is continuous, β(ψn ) converges to φ in the L1 sense. The Riesz Lemma gives a subsequence such that lim β(ψnk )(x) = φ(x) almost everywhere.

k→∞ Applying Lemma 0.3 in a W ss -foliation box is permissable because the foliation is absolutely continuous. Hence φ is essentially constant along strong stable plaques.

4 CHARLES PUGH, MICHAEL SHUB, AND ALEXANDER STARKOV

Covering M with foliation boxes completes the proof that φ is essentially constant along strong stable leaves. Replacing f with f −1 gives the same assertion for the strong unstable foliation.

Finally, if φ is measurable but not L1, we replace it by a cut-off version φ(x) if |φ(x)| ≤ L φL (x) = 0 if |φ(x)| L.

Clearly, φL is L1 and f -invariant. Essential constancy of φL along the leaves of the strong stable and strong unstable foliations implies the same for φ The next lemma generalizes the fact that almost everywhere invariance of a measurable function along orbits of a flow is implied by almost everywhere invariance for each time-t map.

Lemma 0.5.

Suppose that the smooth manifold M carries a smooth measure m, φ : M → R is a measurable function, G is a Lie group that acts smoothly on M,

and the G-orbits foliate M. The following are equivalent:

(a) φ is essentially constant along the orbits of G.

(b) For each g ∈ G, there is a zero set Zg ⊂ M such that for all x ∈ M \ Zg, φ(x) = φ(gx).

(c) There is a single zero set Z such that for all x ∈ M \ Z, and for all g ∈ G, φ(x) = φ(gx).

Proof We write the action of G on M as x → gx, and show that (a) ⇒ (b) ⇒ (c) ⇒ (a).

Assume (a). Then there is a zero set Z ⊂ M such that if x, x ∈ M \ Z belong to a common G-orbit then φ(x) = φ(x ). Fix g ∈ G and define Zg = Z ∪ g −1 Z.

Since G acts smoothly, Zg is a zero set. If x ∈ M \ Zg then x, gx ∈ M \ Z, and by (a), φ(x) = φ(gx). This gives (b).

Assume (b). Uncountability of G precludes taking Z as the union of the Zg, g ∈ G. We first look at the question in a foliation box U ⊂ M, which we coordinatize as W × Y where the local G-orbits are plaques P = W × y. We claim that there is a zero set Z(U ) ⊂ U such that if x, x ∈ U \ Z(U ) lie on a common plaque then φ(x) = φ(x ). Fix any a b and consider the sets A = {x ∈ U : φ(x) ≤ a} and B = {x ∈ U : φ(x) ≥ b}.

Let Z(a, b) be the union of those plaques such that both slices Ay = {w : (w, y) ∈ A} and By = {w : (w, y) ∈ B} have positive W -area. It is enough to show that Z(a, b) is a zero set for then we can take Z(U ) = Z(a, b) where (a, b) ranges over all rationals with a b.

Suppose the local assertion is false: there is a set Y0 ⊂ Y of positive Y -area such that for each y ∈ Y0, both slices Ay and By have positive W -area. Let A0 be the set of density points of A. It differs from A by a zero set. Fubini’s Theorem implies that almost every plaque meets A in a W -zero set if and only if it meets A0 in a W -zero set. The same is true for B. Thus, almost every plaque in W × Y0 contains density points of both A and B. Fix such a pair of density points p, q in a common

UNIQUE ERGODICITY, STABLE ERGODICITY AND THE MAUTNER PHENOMENON FOR DIFFEOMORPHISMS

plaque. Note that p, q need not lie in A, B, and their slices need not have positive W -area.

Since p, q lie in a common plaque we can fix a g ∈ G with gp = q, where we write the action as hg (x) = gx. Since G acts smoothly on M, F : x → gx is a diffeomorphism that slides along plaques and sends p to q. The plaques meeting Zg in sets of non-zero W -area form a zero set, and discarding it from U produces a subset U of full measure such that for every plaque P ⊂ U, φ(gx) = φ(x) almost everywhere with respect to W -area on P.

Thus, for each plaque P ⊂ U, F (A ∩ P ) = A ∩ P modulo a W -zero set on P, where A = A ∩ U. Since A and A differ by a zero set, p is a density point of A.

A diffeomorphism preserves all density points, so F (p) = q is a density point of A, and hence of A. This is obviously impossible – A and B cannot have a common density point.

Globalization presents no problem. For M can be covered by countably many open foliation boxes U, and discarding every G-orbit that meets one of the zero sets Z(U ) leaves a full measure set M ⊂ M, and φ is essentially constant along every G-orbit in M. This gives (c).

Assume (c). Then there is a zero set Z ⊂ M, such that for all g ∈ G and all x ∈ M \ Z, we have φ(x) = φ(gx). Now, if x, x ∈ M \ Z lie on the same G-orbit then x = gx for some g ∈ G, and hence φ(x) = φ(x ), which gives (a).

Lemma 0.6.

If φ : M → R is measurable and h : G → Homeo(M ) is a nice action then the stabilizer St(φ) = {g ∈ G : φ(x) = φ(hg (x)) a.e.} is a closed subgroup of G.

Proof Here, “nice” means that M is locally compact, metrizable, h is continuous, µ is a regular probability measure on M, and the Radon-Nikodym derivatives of hg are locally uniformly bounded. In the case at hand, µ is a G-invariant measure on the homogeneous space M = G/B, and the action is left or right G-multiplication.

Suppose that g, g ∈ St(φ). Absolute continuity implies that hg is a zero-setpreserving change of variables. Hence φ ◦ hg = φ (a.e.) ⇒ φ ◦ hg ◦ hg = φ ◦ hg = φ (a.e).

Since hgg = hg ◦ hg, we have gg ∈ St(φ). Similarly, absolute continuity of hg−1 gives φ ◦ hg = φ (a.e.) ⇒ φ ◦ hg ◦ hg−1 = φ ◦ hg−1 (a.e), −1 and hence g ∈ St(φ), which completes the proof that the stabilizer is a subgroup of G.

To prove closedness, suppose that gn → g and gn ∈ St(φ) for all n. Call hgn = hn and hg = h. We must show that φ ◦ h = φ almost everywhere. Since the issue is local, it is enough to choose a compact neighborhood N of an arbitrary x0 ∈ X and show that φ ◦ h = φ almost everywhere on N. Continuity of the action and compactness of N imply that hn |N → h|N uniformly. Thus there is a compact neighborhood W of h(N ) such that for all n ≥ some n0, we have hn (N ) ⊂ W.

6 CHARLES PUGH, MICHAEL SHUB, AND ALEXANDER STARKOV

–  –  –

Unique ergodocity Due to Dani (see [St1] §1), f ∈ Aff(G/B) is well known to be a K-automorphism of G/B iff G = HB. We prove the following result.

Theorem 0.7.

Let f : G/B → G/B be an affine map on a compact space G/B of finite volume.

Then the following conditions are equivalent:

(1) f is a K-automorphism of G/B, i.e. G = HB, (2) Gs is ergodic on G/B, (3) Gs is minimal on G/B, (4) Gs is strictly ergodic on G/B.

We do not know how to prove Theorem 0.7 directly. Instead, we show how to derive it (at least, in many cases; the general case is considered somewhat differently) from the following classical result. Recall that f is said to be semisimple if df : g → g is diagonalizable over C.

Theorem 0.8.

[B],[V],[EP] Let f : G/B → G/B be a semisimple affine map on a compact space G/B of finite volume. Then the conditions (2), (3), and (4) from

Theorem 1 are equivalent to the following condition:

(1 ) f is weak mixing on G/B.



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