Absolutely singular dynamical foliations

David Ruelle and Amie Wilkinson

September 10, 2002

Introduction

Let A_{2} be the automorphism of the 2-torus, T^{2} = R^{2}/Z^{2}, given by ( \begin{matrix}2 & 1 \\ 1 & 1 \end{matrix} ) .

Let A_{3} be the automorphism of the 3-torus T^{3} = R^{3}/Z^{3} given by ( \begin{matrix}A_{2} & 0 \\ 0 & 1 \end{matrix} ) .

Let Diff_{μ}^{2}(T^{3}) be the set of C^{2} diffeomorphisms of T^{3} that preserve Lebesgue-Haar measure μ.

In [SW1], M. Shub and A. Wilkinson prove the following theorem.

Theorem: Arbitrarily close to A_{3} there is a C^{1}-open set U ⊂ Diff_{μ}^{2}(T^{3}) such that for each g ∈ U,

1. g is ergodic.

2. There is an equivariant fibration π : T^{3} → T^{2} such that πg = A_{2}π The fibers of π are the leaves of a foliation W_{g}^{c} of T^{3} by C^{2} circles. In particular, the set of periodic leaves is dense in T^{3}.

3. There exists λ^{c} > 0 such that, for μ-almost every w ∈ T^{3}, if v ∈ T_{w}T^{3} is tangent to the leaf of W_{g}^{c} containing w, then

lim_{n→∞} \frac{1}{n} log ∥T_{w}g^{n}v∥ = λ^{c}.

4. Consequently, there exists a set S ⊆ T^{3} of full μ-measure that meets every leaf of W_{g}^{c} in a set of leaf-measure 0. The foliation W_{g}^{c} is not absolutely continuous.

Additionally, it is shown that the diffeomorphisms in U are nonuniformly hyperbolic and Bernoullian. In this note, we prove:

Theorem I: Let g satisfy conclusions 1.–3. of the previous theorem. Then there exist S ⊆ T^{3} of full μ-measure and k ∈ N such that S meets every leaf of W_{g}^{c} in exactly k points. The foliation W_{g}^{c} is absolutely singular.

Remark: In A. Katok’s example of an absolutely singular foliation in [Mi], the leaves of the foliation meet the set of full measure in one point. In the [SW1] examples, the set S may necessarily meet leaves of W_{g}^{c} in more than one point, as the following argument of Katok’s shows.

It follows from Theorem II in [SW2] that for k ∈ Z_{+} and for small a, b > 0, the map g = j_{a,k} ∘ h_{b} satisfies the hypotheses of Theorem I, where

h_{b}(x, y, z) = (2x + y, x + y, x + y + z + b sin 2πy), and
j_{a,k}(x, y, z) = (x, y, z) + a cos(2πkz) · (1 + \sqrt{5}, 2, 0).

For k ∈ N, let ρ_{k} be the vertical translation that sends (x, y, z) to (x, y, z+ \frac{1}{k}). Note that h_{b} ∘ ρ_{k} = ρ_{k} ∘ h_{b} and j_{a,k} ∘ ρ_{k} = ρ_{k} ∘ j_{a,k}. Thus g ∘ ρ_{k} = ρ_{k} ∘ g.

The fibration π : T^{3} → T^{2} was obtained in [SW1] by using the persistence of normally hyperbolic submanifolds under perturbations. In the present case the symmetries ρ_{k} preserve the fibers of the trivial fibration P : T^{3} → T^{2} from which one starts, and also the maps g. Therefore the fibers of π : T^{3} → T^{2} (i.e., the leaves of center foliation W_{g}^{c}) are invariant under the action of the finite group < ρ_{k} >.

Let S be the (full measure) set of points in T^{3} for which the center direction is a positive Lyapunov direction (i.e. for which conclusion 3 holds). Since ρ_{k}(W_{g}^{c}) = W_{g}^{c}, it follows that ρ_{k}S = S. If p ∈ S ∩ W^{c}(p), then ρ_{k}(p) ∈ ρ_{k}(S) ∩ ρ_{k}(W^{c}(p)) = S ∩ W^{c}(p); that is, S ∩ W^{c}(p) contains at least k points.

Thus Theorem I is “sharp” in the sense that we cannot say more about the value of k in general. We see no reason why k = 1 should hold even for a residual set in U.

Theorem I has an interesting interpretation. Recall that a G-extension of a dynamical system f : X → X is a map f_{φ} : X × G → X × G, where G is a compact group, of the form (x, y) ↦ (g(x), φ(x)y). If f preserves ν, and φ : X → G is measurable, then f_{φ} preserves the product of ν with Lebesgue-Haar measure on G. A Z/kZ-extension is also called a k-point extension.

Let λ be an invariant probability measure for a k-point extension of f : X → X, and {λ_{x}} the family of conditional measures associated with the partition {{x} × G}. We remark that if λ is ergodic, then each atom of λ_{x} must have the same weight 1/k (up to a set of λ-measure 0).

Now take g ∈ U. Choose a coherent orientation on the leaves of {π^{−1}(x)}_{x∈T^{2}}. Take h : T^{3} → T^{2} × T to be any continuous change of coordinates such that h restricted to π^{−1}(x) is smooth and orientation preserving to {x} × T. We may then write F = h ∘ g ∘ h^{−1} : T^{2} × T → T^{2} × T in the form

F (x, p) = (A_{2}x, φ_{x}(p))

where φ_{x} : T → T is smooth and orientation preserving. If P : T^{2} × T → T^{2} is the projection on the first factor of the product, we have P ∘ h = π. Therefore, writing λ = h^{∗}μ, we have P^{∗}λ = π^{∗}μ. Let {λ_{x}} be the disintegration of the measure λ along the fibers {x} × T. By a further measurable change of coordinates, smooth along each {x} × T fiber, we may assume that λ-almost everywhere, the atoms of λ_{x} are at l/k, for l = 0, …, k − 1. But then φ_{x} permutes the atoms cyclically, and we obtain the following corollary.

Corollary: For every g ∈ U there exists k ∈ N such that (T^{3}, μ, g) is isomorphic to an (ergodic) k-point extension of (T^{2}, π^{∗}μ, A_{2}).

M. Shub has observed that if g = j_{a,k} ∘ h_{b}, then π^{∗}μ is actually Lebesgue measure on T^{2}.

1 Proof of Theorem I

The proof of Theorem I follows from a more general result about fibered diffeomorphisms. Before stating this result, we describe the underlying setup and assumptions.

Let X be a compact metric space with Borel probability measure ν, and let f : X → X be invertible and ergodic with respect to ν. Let M be a closed Riemannian manifold and φ : X → Diff^{1+α}(M) a measurable map. Consider the skew-product transformation F : X × M → X × M given by

F (x, p) = (f(x), φ_{x}(p)).

Assume further that there is an F -invariant ergodic probability measure μ on X × M such that π_{∗}μ = ν, where π : X × M → X is the projection onto the first factor.

For x ∈ X, let φ_{x}^{(0)} be the identity map on M and for k ∈ Z, define φ_{x}^{(k)} by

φ_{x}^{(k+1)} = φ_{f^{k}(x)} ∘ φ_{x}^{(k)}.

Since the tangent bundle to M is measurably trivial, the derivative map of φ along the M direction gives a cocycle Dφ : X × M × Z → GL(n, R), where n = dim(M):

(x, p, k) ↦ D_{p}φ_{x}^{(k)}.

Assume that log^{+} ∥Dφ∥_{α} ∈ L^{1}(X × M, μ), where ∥ · ∥_{α} is the α-Hölder norm. Let λ_{1} < λ_{2} ⋯ < λ_{l} be the Lyapunov exponents of this cocycle; they exist for μ-a.e. (x, p) by Oseledec’s Theorem and are constant by ergodicity. We call these the fiberwise exponents of F. Under the assumptions just described, we have the following result.

Theorem II: Suppose that λ_{l} < 0. Then there exists a set S ⊆ X × M and an integer k ≥ 1 such that

• μ(S) = 1

• For every (x, p) ∈ S, we have #(S ∩ {x} × M) = k.

This has the immediate corollary:

Corollary: Let f ∈ Diff^{1+α}(M). If μ is an ergodic measure with all of its exponents negative, then it is concentrated on the orbit of a periodic sink.

The corollary has a simple proof using regular neighborhoods. Our proof is a fibered version. Theorem I is also a corollary of Theorem II. For this, the argument is actually applied to the inverse of g, which has negative fiberwise exponents, rather than to g itself, whose fiberwise exponents are positive. As we described in the previous remarks, there is a measurable change of coordinates, smooth along the leaves of W_{g}^{c} in which g^{−1} is expressed as a skew product of T^{2} × T.

Remark: Without the assumption that f is invertible, Theorem II is false. An example is described by Y. Kifer [Ki], which we recall here. Let f : T → T be a C^{1+α} diffeomorphism with exactly two fixed points, one attracting and one repelling. Consider the following random diffeomophism of T: with probability p ∈ (0, 1), apply f, and with probability 1 − p, rotate by an angle chosen randomly from the interval [−ϵ, ϵ].

Let X = ({0, 1} × T)^{N}. To generate a sequence of diffeomorphisms f_{0}, f_{1}, … according to the above rule, we first define φ : X → Diff^{1+α}(T) by

φ(ω) = \{ \begin{matrix}f & if ω(0) = (0, θ), \\ R_{θ} & if ω(0) = (1, θ), \end{matrix}

where R_{θ} is rotation through angle θ. Next, we let ν_{ϵ} be the product of p, 1−p-measure on {0, 1} with the measure on T that is uniformly distributed on [−ϵ, ϵ]. Then corresponding to ν_{ϵ}^{N}-almost every element ω ∈ X is the sequence {f_{k} = φ(σ^{k}(ω))}_{k=0}^{∞}, where σ : X → X is the one-sided shift σ(ω)(n) = ω(n + 1).

Put another way, the random diffeomorphism is generated by the (noninvertible) skew product τ : X ×T → X ×T, where τ(ω, x) = (σ(ω), φ(ω)(x)). An ergodic ν_{ϵ}-stationary measure for this random diffeomorphism is a measure μ_{ϵ} on T such that μ_{ϵ} × ν_{ϵ}^{N} is τ-invariant and ergodic. Such measures always exist ([Ki], Lemma I.2.2), but, for this example, there is an ergodic stationary measure with additional special properties.

Specifically, for every ϵ > 0, there exists an ergodic ν_{ϵ}-stationary measure μ_{ϵ} on T such that, as ϵ → 0, μ_{ϵ} → δ_{x_{0}}, in the weak topology, where δ_{x_{0}} is Dirac measure concentrated on the sink x_{0} for f. From this, it follows that, as ϵ → 0, the fiberwise Lyapunov exponent for μ_{ϵ} approaches log |f′(x_{0})| < 0, which is the Lyapunov exponent of δ_{x_{0}}. Thus, for ϵ sufficiently small, the fiberwise exponent for τ with respect to μ_{ϵ} is negative. Nonetheless, it is easy to see that μ_{ϵ} for ϵ > 0 cannot be uniformly distributed on k atoms; if μ_{ϵ} were atomic, then τ-invariance of μ_{ϵ} × ν_{ϵ}^{N} would imply that, for every x ∈ T,

μ_{ϵ}({x}) = pμ_{ϵ}({f^{−1}(x)}) + (1 − p) ∫_{−ϵ}^{ϵ} μ_{ϵ}({R_{θ}(x)})dθ
= pμ_{ϵ}({f^{−1}(x)}),

which is impossible if μ_{ϵ} has finitely many atoms. In fact, μ_{ϵ} can be shown to be absolutely continuous with respect to Lebesgue measure (see [Ki], p. 173ff and the references cited therein). Hence invertibility is essential, and we indicate in the proof of Theorem II where it is used.

Proof of Theorem II: We first establish the existence of fiberwise “stable manifolds” for the skew product F. A general theory of stable manifolds for random dynamical systems is worked out in ([Ki], Theorem V.1.6; see also [BL]); since we are assuming that all of the fiberwise exponents for F are negative, we are faced with the simpler task of constructing fiberwise regular neighborhoods for F (see the Appendix by Katok and Mendoza in [KH]). We outline a proof, following closely [KH].

Theorem 1.1 (Existence of Regular Neighborhoods) There exists a set Λ_{0} ⊆ X × M of full measure such that for ϵ > 0:

• There exists a measurable function r : Λ_{0} → (0, 1] and a collection of embeddings Ψ_{(x,p)} : B(0, q(x, p)) → M such that Ψ_{(x,p)}(0) = p and exp(−ϵ) < r(F (x, p))/r(x, p) < exp(ϵ).

• If φ_{(x,p)} = Ψ_{F (x,p)}^{−1} ∘ φ_{x} ∘ Ψ_{(x,p)} : B(0, r(x, p)) → R^{n}, then D_{0}φ_{(x,p)} satisfies

exp(λ_{1} − ϵ) ≤ ∥D_{0}φ_{(x,p)}^{−1}∥^{−1}, ∥D_{0}φ_{(x,p)}∥ ≤ exp(λ_{l} + ϵ).

• The C^{1} distance d_{C^{1}}(φ_{(x,p)}, D_{0}φ_{(x,p)}) < ϵ in B(0, r(x, p)).

• There exist a constant K > 0 and a measurable function A : Λ_{0} → R such that for y, z ∈ B(0, r(x, p)),

K^{−1}d(Ψ_{(x,p)}(y), Ψ_{(x,p)}(z)) ≤ ∥y − z∥ ≤ A(x)d(Ψ_{(x,p)}(y), Ψ_{(x,p)}(z)),

with exp(−ϵ) < A(F (x, p))/A(x, p) < exp(ϵ).

Proof: See the proof of Theorem S.3.1 in [KH]. □

Decompose μ into a system of fiberwise measures dμ(x, p) = dμ_{x}(p)dν(x). Invariance of μ with respect to F implies that, for ν-a.e. x ∈ X,

φ_{x}_{∗}μ_{x} = μ_{f(x)}.

Corollary 1.2 There exists a set Λ ⊆ X × M, and real numbers R > 0, C > 0, and c < 1 such that

(1) μ(Λ) > .5, and, if (x, p) ∈ Λ, then μ_{x}(Λ_{x}) > .5, where Λ_{x} = {p ∈ M | (x, p) ∈ Λ},

(2) If (x, p) ∈ Λ and d_{M}(p, q) ≤ R, then

d_{M}(φ_{x}^{(m)}(p), φ_{x}^{(m)}(q)) ≤ Cc^{m}d_{M}(p, q),

for all m ≥ 0.

Proof: This follows in a standard way from the Mean Value Theorem and Lusin’s Theorem.□

To prove Theorem II, it suffices to show that there is a positive ν-measure set B ⊆ X, such that for x ∈ B, the measure μ_{x} has an atom, as the following argument shows. For x ∈ X, let d(x) = sup_{p∈M} μ_{x}(p). Clearly d is measurable, f-invariant, and positive on B. Ergodicity of f implies that d(x) = d > 0 is positive and constant for almost all x ∈ X. Let S = {(x, p) ∈ X × M | μ_{x}(p) ≥ d}. Observe that S is F -invariant, has measure at least d, and hence has measure 1. The conclusions of Theorem II follow immediately.

Let Λ, R > 0, C > 0, and c < 1 be given by Corollary 1.2, and let B = π(Λ). Let N be the number of R/10-balls needed to cover M. We now show that for ν-almost every x ∈ B, the measure μ_{x} has at least one atom.

For x ∈ X, let

m(x) = inf ∑ diam (U_{j}),

where the infimum is taken over all collections of closed balls U_{1}, …, U_{k} in M such that k ≤ N and μ_{x}(⋃_{j=1}^{k} U_{j}) ≥ .5. Let m = ess sup _{x∈B}m(x).

We now show that m = 0. If m > 0, then there exists an integer J such that

CΔc^{J}N < m/2,  (1)

where Δ is the diameter of M. Let U be a cover of M by N closed balls of radius R/10. For x ∈ B, let U_{1}(x), …, U_{k(x)}(x) be those balls in U that meet Λ_{x}. Since these balls cover Λ_{x}, and μ_{x}(Λ_{x}) > .5, it follows that μ_{x}(⋃_{j=1}^{k(x)} U_{j}(x)) ≥ .5. But φ_{x}^{(i)}_{∗}μ_{x} = μ_{f^{i}(x)}, and so it’s also true that

μ_{f^{i}(x)}( ⋃_{j=1}^{k(x)} φ_{x}^{(i)}(U_{j}(x))) ≥ .5,  (2)

for all i.

We now use the fact that φ_{x}^{(i)} contracts regular neighborhoods to derive a contradiction. The balls U_{j}(x) meet Λ_{x} and have diameter less than R/10, and so by Corollary 1.2, (2), we have

diam (φ_{x}^{(i)}(U_{j}(x))) ≤ CΔc^{i}.  (3)

Let τ : B → N be the first-return time of f^{J} to B, so that f^{Jτ(x)}(x) ∈ B, and f^{Ji}(x) ∉ B, for i ∈ {1, …, τ(x) − 1}. Decompose the set B according to these first return times:

B = ⋃_{i=1}^{∞} B_{i} (mod 0),

where B_{i} = τ^{−1}(i). Because f is invertible and f^{−1} preserves measure, we also have the mod 0 equivalence:

B′ := ⋃_{i=1}^{∞} f^{Ji}(B_{i}) = B (mod 0).

Let y ∈ B′. Then y = f^{Ji}(x), where x ∈ B_{i} ⊆ B, for some i ≥ 1. It follows from the definition of m(y) and inequalities (2), (3) and (1) that

m(y) ≤ ∑_{j=1}^{k(x)} diam (φ_{x}^{(Ji)}(U_{j}(x)))
≤ Ck(x)Δc^{Ji}
≤ CNΔc^{J}
< m/2.

But then

m = ess sup _{x∈B}m(x)
= ess sup _{y∈B′}m(y)
< m/2,

contradicting the assumption m > 0.

Thus m = 0, and, for ν-almost every x ∈ B, we have m(x) = 0. If m(x) = 0, then there is a sequence of closed balls U^{1}(x), U^{2}(x), ⋯ with lim_{i→∞} diam (U^{i}(x)) = 0 and μ_{x}(U^{i}(x)) ≥ .5/N, for all i. Take p_{i} ∈ U^{i}(x); any accumulation point of {p_{i}} is an atom for μ_{x}. Since we have shown that μ_{x} has an atom, for ν-a.e. x ∈ B, the proof of Theorem II is complete.□

We thank Michael Shub and Anatole Katok for useful conversations.

References

[BL] J. Bahnmüller and P.-D. Liu. “Characterization of measures satisfying Pesin’s entropy formula for random dynamical systems.” J. Dynam. Diff. Equ. 10 (1998), no. 3, 425-448.

[KH] Katok, A. and B. Hasselblatt, “Introduction to the modern theory of dynamical systems.” Cambridge, 1995.

[Ki] Kifer, Y. “Ergodic theory of random transformations,” Birkhäuser Boston, 1986.

[Mi] Milnor, J. Fubini foiled: Katok’s paradoxical example in measure theory. Math. Intelligencer, 19 (1997), no. 2, 30-32.

[SW1] Shub, M. and Wilkinson, A., Pathological foliations and removable zero exponents, Inv. Math., to appear.

[SW2] Shub, M. and Wilkinson, A., A stably Bernoullian diffeomorphism that is not Anosov, preprint.