Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 489–500

PACKING DIMENSIONS, TRANSVERSAL MAPPINGS AND GEODESIC FLOWS

Mika Leikas

University of Jyväskylä, Department of Mathematics and Statistics P.O. Box 35, FI-40014 Finland; mileikas@maths.jyu.fi

Abstract. In this work we first generalize the projection result by K. Falconer and J. Howroyd concerning packing dimensions of projected measures on R^{n} to parametrized families of transversal mappings between smooth manifolds and measures on them. After this we compute the packing dimension of the natural projection of a probability measure which is invariant under the geodesic flow on the unit tangent bundle of a two-dimensional Riemannian manifold.

1. Introduction

The behavior of the Hausdorff dimension, dim_{H} , under projection-type mappings is well known. In the 1950’s Marstrand [Mar] proved that the Hausdorff dimension of a planar set is preserved under typical orthogonal projections. In [K] Kaufman verified the same result using potential theoretic methods, and in [Mat1] Mattila generalized it to higher dimensions. For measures the analogous principle, discovered by Kaufman [K], Mattila [Mat2], Hu and Taylor [HT], and Falconer and Mattila [FM], can be formulated in the following way: If μ is a compactly supported Radon measure on R^{n}, then for almost all V ∈ G(n, m)

dim_{H} P_{V} μ = dim_{H} μ provided that dim_{H} μ ≤ m.

On the other hand, if dim_{H} μ > m, then

P_{V} μ ≪ ℋ ^{m}|_{V}

for almost all V ∈ G(n, m). In addition, if the m-energy of μ is finite, then

P_{V} μ ≪ ℋ ^{m}|_{V} with the Radon–Nikodym derivative in L^{2}(V, ℋ ^{m}|_{V})

for almost all V ∈ G(n, m). Here G(n, m) is the Grassmann manifold of all mdimensional linear subspaces of R^{n}, P_{V}: R^{n} → V is the orthogonal projection onto V ∈ G(n, m), and P_{V} μ is the image of μ under P_{V} defined by the formula P_{V} μ(A) = μ(P_{V}^{−1}A) for A ⊂ V . Moreover, by μ ≪ ν we denote the absolute continuity of a measure μ with respect to a measure ν, and μ|_{A} is the restriction of a measure μ to a set A, that is μ|_{A}(B) = μ(A ∩ B).

2000 Mathematics Subject Classification: Primary 58C35, 53D25, 28A78, 37D40, 37A05, 28A80.

The author was supported by the Graduate School of Mathematical Analysis and its applications, and by the foundation Vilho, Yrjö ja Kalle Väisälän rahasto.

In [FH], Falconer and Howroyd proved an analogous result for the packing dimension, dim_{p} , of projected measures. They showed that if μ is a finite Borel measure on R^{n}, then

(1.1)  dim_{p} P_{V} μ = dim_{m} μ

for almost all V ∈ G(n, m), where dim_{m} μ is a packing-type dimension defined by using a certain m-dimensional kernel. This result tells that the packing dimension is the same for almost all projections, but it may happen that dim_{m} μ < dim_{p} μ.

The above results are “almost all”-results giving no information about any specific projection. However, as discovered by Ledrappier and Lindenstrauss [LL], similar methods work for one particular projection. In [LL] they studied measures on the unit tangent bundle SM of a compact two-dimensional Riemannian manifold M. They showed that, if μ is a probability measure on SM and μ is invariant under the geodesic flow, then

(1) dim_{H} Πμ = dim_{H} μ, if dim_{H} μ ≤ 2, and

(2) Πμ ≪ ℋ ^{2}|_{M}, if dim_{H} μ > 2.

Here Π: SM → M is the natural projection. Ledrappier and Lindenstrauss proved also that the Radon–Nikodym derivative of Πμ is an L^{2} -function, if μ has finite α-energy for some α > 2.

Inspired by the results in [LL], in [JJLe] we reproved the above theorem and showed that if the α-energy of μ is finite for some α > 2, then Πμ has fractional derivatives of order γ in L^{2} for all γ < \frac{1}{2}(α − 2). To achieve this new proof we used the generalized projection formalism introduced by Peres and Schlag in [PS]. Our proof also explains why this kind of projection result fails when the dimension of the base manifold is greater than two.

In this paper we consider the natural question of how the packing dimension of an invariant measure behaves under the natural projection. To achieve this, in Section 3 we first generalize (1.1) to parametrized families of transversal mappings between manifolds (Theorem 3.6). The methods of Falconer and Howroyd do not directly work in our setting, but circumventing some technical problems eventually leads to a bit easier proof. After this, in Section 4 we compute the packing dimension of Πμ, when the setting is similar to that in [LL] (Theorem 4.2). Finally, in Section 5 we show that, unlike the Hausdorff dimension, the packing dimension of a (locally) invariant measure may decrease under the projection even in the two-dimensional case.

2. Preliminaries and definitions

In the following definitions (X, d_{X}) is a metric space, and B(x, r) is the open ball with center at x ∈ X and radius r > 0.

Definition 2.1. Let μ be a finite Borel measure on X. The (lower) packing dimension of μ is

dim_{p} μ := sup{t ≥ 0 : lim inf_{r→0} r^{−t}μ(B(x, r)) = 0 for μ-a.a. x ∈ X}

= inf{dim_{p} A : A ⊂ X is a Borel set with μ(A) > 0}

= μ- ess inf_{x∈X}(lim sup_{r→0} \frac{log μ(B(x, r))}{log r}),

and the upper packing dimension of μ is

dim_{p}^{∗} μ := inf{t > 0 : lim inf_{r→0} r^{−t}μ(B(x, r)) > 0 for μ-a.a. x ∈ X}

= inf{dim_{p} A : A ⊂ X is a Borel set with μ(X \ A) = 0}

= μ- ess sup_{x∈X}(lim sup_{r→0} \frac{log μ(B(x, r))}{log r}).

As in [FH], for every k ∈ N we define a new dimension of a measure, which may be regarded as a packing-type dimension defined in terms of a certain k - dimensional kernel. For more properties of this k -dimension, see [FH].

Definition 2.2. Let μ be a finite Borel measure on X. For every x ∈ X, r > 0 and k ∈ N, we define

F_{k}^{μ}(x, r) := ∫_{X} min\{1, r^{k}d_{X}(x, y)^{−k}\} dμ(y) = kr^{k} ∫_{r}^{∞} \frac{μ(B(x, h))}{h^{k+1}} dh,

where the last equality follows directly from Fubini’s theorem. Furthermore, we define

dim_{k} μ := sup{t ≥ 0 : lim inf_{r→0} r^{−t}F_{k}^{μ}(x, r) = 0 for μ-a.a. x ∈ X}

and

dim_{k}^{∗} μ := inf{t > 0 : lim inf_{r→0} r^{−t}F_{k}^{μ}(x, r) > 0 for μ-a.a. x ∈ X}

for all k ∈ N.

The following theorem is from [FH, Corollary 3].

Theorem 2.3. Let ν be a finite Borel measure on R^{n}. Then

dim_{p} ν = dim_{n} ν  and  dim_{p}^{∗} ν = dim_{n}^{∗} ν.

Falconer and Howroyd proved also the following [FH, Theorem 6]:

Theorem 2.4. Let ν be a finite Borel measure on R^{n}. Then

dim_{p} P_{V} ν = dim_{m} ν  and  dim_{p}^{∗} P_{V} ν = dim_{m}^{∗} ν

for γ_{n,m} -almost all V ∈ G(n, m), where γ_{n,m} is the unique orthogonally invariant probability measure on G(n, m).

3. Packing dimensions and transversal mappings between manifolds

In this section we generalize Theorem 2.4 to families of transversal mappings. First we notice that Theorem 2.3 holds also on manifolds.

Theorem 3.1. Let (N, d_{N}) be a smooth n-dimensional Riemannian manifold equipped with the distance function d_{N} induced by the Riemannian metric, and let μ be a finite, compactly supported Borel measure on N. Then

dim_{p} μ = dim_{n} μ  and  dim_{p}^{∗} μ = dim_{n}^{∗} μ.

Proof. The proof of this theorem is the same as the proof of Theorem 2.3 [FH, Corollary 3]. The lemmas they need from [FM] are true also in our setting.

For the generalization of Theorem 2.4 we first prove a lemma which gives the natural upper bound for the packing dimension of the image measure whenever the mapping is Lipschitz continuous.

Lemma 3.2. Let (N, d_{N}) and (M, d_{M}) be smooth Riemannian manifolds with dimensions n and m, respectively. Let P: N → M be a Lipschitz continuous mapping, and let μ be a finite Borel measure on N. Then

dim_{m} P μ ≤ dim_{m} μ  and  dim_{m}^{∗} P μ ≤ dim_{m}^{∗} μ.

Proof. We denote C := Lip(P) + 1 ≥ 1. Let 0 ≤ t < dim_{m} P μ, and define

M(t) := {u ∈ M : lim inf_{r→0} r^{−t}F_{m}^{P μ}(u, r) = 0}.

First we notice that by standard arguments u ↦ lim inf_{r→0} r^{−t}F_{m}^{P μ}(u, r) is a Borel function and thus M(t) is a Borel set. From the definition of dim_{m} P μ we get that μ(N \P^{−1}(M(t))) = P μ(M \M(t)) = 0. Now take x ∈ P^{−1}(M(t)) ⊂ N and denote u := P (x) ∈ M(t). Then

lim inf_{r→0} r^{−t}F_{m}^{μ}(x, r) = lim inf_{r→0} r^{−t} ∫_{N} min{1, r^{m}d_{N}(x, y)^{−m}} dμ(y)

≤^{(∗)} C^{m} lim inf_{r→0} r^{−t} ∫_{N} min{1, r^{m}d_{M}(P (x), P (y))^{−m}} dμ(y)

= C^{m} lim inf_{r→0} r^{−t} ∫_{P (N)} min{1, r^{m}d_{M}(P (x), v)^{−m}} dP μ(v)

=^{(∗∗)} C^{m} lim inf_{r→0} r^{−t} ∫_{M} min{1, r^{m}d_{M}(u, v)^{−m}} dP μ(v)

= C^{m} lim inf_{r→0} r^{−t}F_{m}^{P μ}(u, r) = 0,

where (∗) follows from the fact that d_{M}(P (x), P (y)) ≤ Cd_{N}(x, y) and (∗∗) holds because P μ(M \ P (N)) = 0. Hence t ≤ dim_{m} μ, and so dim_{m} P μ ≤ dim_{m} μ proving the first inequality. Next take 0 ≤ t < dim_{m}^{∗} P μ. Again by definition μ(P^{−1}(M(t))) = P μ(M(t)) > 0. The same calculation as above shows that

lim inf_{r→0} r^{−t}F_{m}^{μ}(x, r) = 0

for all x ∈ P^{−1}(M(t)) and thus t ≤ dim_{m}^{∗} μ. This proves the claim.

Next we introduce the setting we are going to work with. Let (L, d_{L}) be a smooth, bounded l-dimensional Riemannian manifold equipped with the distance function d_{L} induced by the Riemannian metric, let (N, d_{N}) be a smooth n-dimensional Riemannian manifold, and let (M, d_{M}) be a smooth m-dimensional Riemannian manifold. We suppose that l, n ≥ m so that in our setting L corresponds to G(n, m), and furthermore, N and M correspond to R^{n} and R^{m} ≅ V ∈ G(n, m), respectively. Let P: L × N → M be a continuous function such that for all j ∈ {0, 1, …} there exists a constant C_{j} such that whenever k_{1} + ⋯ + k_{l} = j,

∥∂_{λ_{1}}^{k_{1}} ⋯ ∂_{λ_{l}}^{k_{l}}P (λ, x)∥ ≤ C_{j}

for all (λ, x) ∈ L × N. Later on we will use the notation P_{λ}(x) := P (λ, x). The basic assumptions we need are the following:

(1) There are finite collections {ϕ, V } and {φ, U} of charts on L and M, respectively, with the following property: there exists R > 0 such that for all λ ∈ L and u ∈ M

B(λ, R) ⊂ V  and  B(u, R) ⊂ U

for some V and U.

(2) The Lipschitz constants of the mappings φ, φ^{−1} , ϕ, and ϕ^{−1} are uniformly bounded from above by a positive constant K.

(3) Mapping T: {(x, y, λ) ∈ N^{2} × L : x ≠ y, d_{M}(P_{λ}(x), P_{λ}(y)) ≤ R} → R^{m},

T_{x,y}(λ) = \frac{φ ∘ P_{λ}(x) − φ ∘ P_{λ}(y)}{d_{N}(x, y)}

is transversal, i.e., there exists a constant C_{T} > 0 such that

det(DT_{x,y}(λ)(DT_{x,y}(λ))^{T} ) ≥ C_{T}^{2} whenever |T_{x,y}(λ)| ≤ C_{T} .

We may refer to this property also by saying that P is transversal. Here A^{T} stands for the transpose of a matrix A. Moreover, we assume that

|∂_{λ_{j}}∂_{λ_{k}}(T_{x,y})_{i}(λ)| ≤ L < ∞

for all j, k ∈ {1, …, l}, i ∈ {1, …, m}, x, y ∈ N, and λ ∈ L.

Throughout the rest of this section the manifolds (L, d_{L}), (N, d_{N}) and (M, d_{M}), and the mapping P: L × N → M will satisfy the above assumptions.

Before stating the main result of this chapter we prove three short lemmas which will help us in the proof of Theorem 3.6. Analogous lemmas can be found in [FH], but since our setting is a bit more general, we have to modify their methods. In particular, in our case we do not have the lower bound for the ℋ ^{l} - measure of the set of exceptional parameters λ (see inequality (3.1)). In fact, circumventing this problem leads to a little bit simpler proof than the one in [FH].

Lemma 3.3. Let (L, d_{L}), (N, d_{N}), (M, d_{M}), and P: L × N → M be as above, and let μ be a finite, compactly supported Borel measure on N. In addition, assume that P is transversal. Then there exist constants C′ > 0 and r_{0} > 0 such that

C′F_{m}^{μ}(x, r) ≥ ∫_{L} μ_{λ}(B(P_{λ}(x), r)) dℋ ^{l}(λ)

whenever 0 < r < r_{0} . Here μ_{λ} := P_{λ}μ.

Proof. We know that under our assumptions there exist constants C′ > 0 and r_{0} > 0 such that

(3.1)  ℋ ^{l}{λ ∈ L : d_{M}(P_{λ}(x), P_{λ}(y)) ≤ r} ≤ C′ min{1, r^{m}d_{N}(x, y)^{−m}}

for all 0 < r < r_{0} and distinct x, y ∈ N (see Lemma 2.1 in [JJN] for the proof). Hence we see by Fubini’s theorem that

C′F_{m}^{μ}(x, r) ≥ ∫_{N} ℋ ^{l}{λ ∈ L : d_{M}(P_{λ}(x), P_{λ}(y)) ≤ r} dμ(y)
= ∫_{L} μ{y ∈ N : d_{M}(P_{λ}(x), P_{λ}(y)) ≤ r} dℋ ^{l}(λ)
= ∫_{L} μ(P_{λ}^{−1}(B(P_{λ}(x), r))) dℋ ^{l}(λ)
= ∫_{L} μ_{λ}(B(P_{λ}(x), r)) dℋ ^{l}(λ)

whenever 0 < r < r_{0} .

Lemma 3.4. Let (L, d_{L}), (N, d_{N}), (M, d_{M}), and P: L × N → M be as above, and let μ be a finite, compactly supported Borel measure on N. In addition, assume that P is transversal. If lim inf_{r→0} r^{−t}F_{m}^{μ}(x, r) = 0, then

lim inf_{r→0} r^{−t}μ_{λ}(B(P_{λ}(x), r)) = 0

for ℋ ^{l} -almost all λ ∈ L, where μ_{λ} = P_{λ}μ.

Proof. Firstly, because

C′F_{m}^{μ}(x, r) ≥ ∫_{L} μ_{λ}(B(P_{λ}(x), r)) dℋ ^{l}(λ)

by Lemma 3.3, we have that

ℋ ^{l}{λ ∈ L : μ_{λ}(B(P_{λ}(x), r)) > kr^{t}} ≤ k^{−1}r^{−t}C′F_{m}^{μ}(x, r)

for all r, k > 0. Now fix k > 0 and choose a strictly decreasing sequence r_{j} → 0 such that

lim_{j→∞} r_{j}^{−t}F_{m}^{μ}(x, r_{j}) = 0.

Since

ℋ ^{l}{λ ∈ L : lim inf_{r→0} r^{−t}μ_{λ}(B(P_{λ}(x), r)) > k}

= ℋ ^{l}( ⋃_{j=1}^{∞}{λ ∈ L : r^{−t}μ_{λ}(B(P_{λ}(x), r)) > k for all r ≤ r_{j}})

= lim_{j→∞} ℋ ^{l}{λ ∈ L : r^{−t}μ_{λ}(B(P_{λ}(x), r)) > k for all r ≤ r_{j}}

≤ lim sup_{j→∞} ℋ ^{l}{λ ∈ L : r_{j}^{−t}μ_{λ}(B(P_{λ}(x), r_{j})) > k}

≤ lim sup_{j→∞} k^{−1}r_{j}^{−t}C′F_{m}^{μ}(x, r_{j}) = 0,

we get that for all k > 0

lim inf_{r→0} r^{−t}μ_{λ}(B(P_{λ}(x), r)) ≤ k

for ℋ ^{l} -almost all λ ∈ L. Thus the claim follows when we let k tend to 0 through a countable sequence.

Lemma 3.5. Let (L, d_{L}), (N, d_{N}), (M, d_{M}), and P: L × N → M be as above, and let μ be a finite, compactly supported Borel measure on N. In addition, assume that P is transversal. For all t ≥ 0 we define

E(t) = {x ∈ N : lim inf_{r→0} r^{−t}F_{m}^{μ}(x, r) = 0}.

(a) If μ(N \ E(t)) = 0, then dim_{p} μ_{λ} ≥ t for ℋ ^{l} -almost all λ ∈ L.

(b) If μ(E(t)) > 0, then dim_{p}^{∗} μ_{λ} ≥ t for ℋ ^{l} -almost all λ ∈ L.

Proof. As in Lemma 3.2 we see by standard methods that E(t) is a Borel set. The proof of (a) is similar to that of Proposition 5(a) in [FH]. For the reader’s convenience we will prove claim (b). By Lemma 3.4 for all x ∈ E(t)

lim inf_{r→0} r^{−t}μ_{λ}(B(P_{λ}(x), r)) = 0

for ℋ ^{l} -almost all λ ∈ L. Therefore Fubini’s theorem implies that for ℋ ^{l} -almost all λ ∈ L

lim inf_{r→0} r^{−t}μ_{λ}(B(u, r)) = 0

for μ_{λ} -almost all u ∈ P_{λ}(E(t)) ⊂ M. Because

μ_{λ}(P_{λ}(E(t))) ≥ μ(E(t)) > 0,

we have that for ℋ ^{l} -almost all λ ∈ L

μ_{λ}({u ∈ M : lim inf_{r→0} r^{−t}μ_{λ}(B(u, r)) = 0}) > 0,

and so dim_{p}^{∗} μ_{λ} ≥ t for ℋ ^{l} -almost all λ ∈ L proving (b).

After these lemmas we get the main theorem almost for free.

Theorem 3.6. Let (L, d_{L}), (N, d_{N}), (M, d_{M}), and P: L × N → M be as above, and let μ be a finite, compactly supported Borel measure on N. In addition, assume that P is transversal. Then

dim_{p} μ_{λ} = dim_{m} μ  and  dim_{p}^{∗} μ_{λ} = dim_{m}^{∗} μ

for ℋ ^{l} -almost all λ ∈ L, where μ_{λ} = P_{λ}μ is the projected measure.

Proof. From Theorem 3.1 and Lemma 3.2 we immediately obtain that dim_{p} μ_{λ} ≤ dim_{m} μ and dim_{p}^{∗} μ_{λ} ≤ dim_{m}^{∗} μ for all λ ∈ L. Hence it is enough to show that the reverse inequalities hold for ℋ ^{l} -almost all λ ∈ L. Assume that 0 < t < dim_{m} μ. Then

lim inf_{r→0} r^{−t}F_{m}^{μ}(x, r) = 0

for μ-almost all x ∈ N. By Lemma 3.5(a) dim_{p} μ_{λ} ≥ t for ℋ ^{l} -almost all λ ∈ L. Taking a countable sequence of t → dim_{p} μ gives the first claim. Assume next that 0 < t < dim_{m}^{∗} μ. Then

lim inf_{r→0} r^{−t}F_{m}^{μ}(x, r) = 0

in a set of points x of positive μ-measure. By Lemma 3.5(b) dim_{p}^{∗} μ_{λ} ≥ t for ℋ ^{l} - almost all λ ∈ L. As above, this gives that ℋ ^{l}{λ ∈ L : dim μ_{λ} < dim_{m} μ} = 0, proving the claim.

Remark 3.7. In all the theorems and lemmas in Section 3 it is possible to replace (N, d_{N}) by a closure of an open and bounded subset of such a manifold, since essentially the only property we need is that the space is n-dimensional.

4. The packing dimension of the projection of a measure invariant under the geodesic flow

In this section we will move from “almost all”-results to the study of one specific projection. The behavior of the Hausdorff dimension of a locally invariant probability measure on the unit tangent bundle of a Riemannian surface under the natural projection was studied in [LL] and [JJLe]. Those results tell us that the Hausdorff dimension is preserved under the natural projection. Next we prove an analogous theorem for the packing dimension of the projected measure. But first we recall the definitions of the geodesic flow and the invariance of a measure under the flow.

Definition 4.1. Let SM be the unit tangent bundle of a smooth, compact Riemannian manifold M. For a given t ∈ R, the geodesic flow F_{t}: SM → SM is a diffeomorphism defined by the condition

F_{t}(x, v) = (γ_{(x,v)}(t), γ′_{(x,v)}(t)),

where γ_{(x,v)} is the unique geodesic with initial conditions γ_{(x,v)}(0) = x and γ′_{(x,v)}(0) = v for every (x, v) ∈ SM.

A measure μ on SM is invariant under the geodesic flow, if F_{t}μ = μ for every t ∈ R, that is μ(F_{t}^{−1}(A)) = μ(A) for A ⊂ SM.

The main result of this section is the following:

Theorem 4.2. Let M be a smooth, compact Riemannian surface, let μ be a Radon probability measure on the unit tangent bundle SM, and let Π: SM → M be the natural projection. If μ is invariant under the geodesic flow, then

dim_{p} Πμ = dim_{2} μ.

Before going into the proof we fix our notation and prove some technical lemmas. For further details see Section 3 in [JJLe].

The invariance of the measure μ implies that

μ = ∑_{j=1}^{K} μ_{j},

where μ_{j} = ψ(ν_{j} × ℒ^{1}), ψ is a bi-Lipschitz mapping from a compact set Ĩ ⊂ R^{3} (basically Ĩ = [0, 1]^{3} ) to its image ψ(Ĩ) ⊂ SM, and ν_{j} is a probability measure on [0, 1]^{2} for every j ∈ {1, …, K}. We call this kind of measures μ_{j} locally invariant. Since

Π(∑_{j=1}^{K} μ_{j}) = ∑_{j=1}^{K} Πμ_{j}  and  dim_{p}(∑_{j=1}^{K} μ_{j}) = min_{1≤j≤K} dim_{p} μ_{j},

it is enough to prove Theorem 4.2 for measures μ_{j}. So we fix some j ∈ {1, …, K} and denote μ̃ := μ_{j} and ν := ν_{j}. In [JJLe] we proved that there exists a transversal mapping P: [0, 1] × [0, 1]^{2} → R, (t, x) ↦ P (t, x) =: P_{t}(x) such that the dimensional behavior of (Φ ∘ Π)μ̃ is similar to that of the measure μ′ defined by the condition

∫_{R^{2}} f(x, t) dμ′(x, t) = ∫_{0}^{1} ∫_{R} f(x, t) dP_{t}ν(x) dℒ^{1}(t)

for all non-negative Borel functions f: R × [0, 1] → [0, ∞]. Here Φ is a chart defined on an open set U ⊂ M such that [0, 1]^{2} ⊂ Φ(U).

Next we prove a lemma which gives us the desired lower bound for the packing dimension of Πμ̃.

Lemma 4.3. Suppose that for all t ∈ [0, 1] we have a compactly supported Radon measure ν_{t} on R. Suppose that μ is a Radon measure on R × [0, 1] such that

∫_{R^{2}} f(x, t) dμ(x, t) = ∫_{0}^{1} ∫_{R} f(x, t) dν_{t}(x) dℒ^{1}(t)

for all non-negative Borel functions f: R × [0, 1] → R. Then, if dim_{p} ν_{t} ≥ α for ℒ^{1} -almost all t ∈ [0, 1], we have that dim_{p} μ ≥ α + 1.

Proof. The proof of this theorem is similar to the proof of the corresponding result for the Hausdorff dimension, which can be found from [JJL, Lemma 3.4]. One essential part of the proof is the fact that since P_{V^{⊥}}μ ≪ ℋ ^{1}|_{V^{⊥}} (see Lemma 3.4 in [JJL]),

(4.1)  ℋ ^{1}- ess inf_{a∈V^{⊥}}{dim_{p} μ_{V,a} : μ_{V,a} ≠ 0} ≤ dim_{p} μ − 1,

where V = x-axis, V^{⊥} is the orthogonal complement of V , and μ_{V,a} is the slice of measure μ by the affine subspace V_{a} = V + a, a ∈ V^{⊥}. For the definition of these slices see [Mat3, Chapter 10]. Inequality (4.1) in turn follows basically from the definition of the packing dimension and from Lemma 5.1 in [F], which says that for every A ⊂ R^{n} and subspace V ∈ G(n, n − m)

dim_{p}(A ∩ V_{a}) ≤ max{dim_{p} A − m, 0}

for ℋ ^{m} -almost all a ∈ V^{⊥}.

Lemma 4.4. We have

dim_{p}(Φ ∘ Π)μ̃ = dim_{2} μ̃

Proof. First we notice that

(4.2)  dim_{2} μ̃ = dim_{1} ν + 1.

Namely, since ψ is a bi-Lipschitz mapping, we have that

F_{1}^{ν}(y, r) = r ∫_{r}^{∞} \frac{ν(B(y, h))}{h^{2}} dh = r ∫_{r}^{∞} \frac{ν(B(y, h)) · h}{h^{3}} dh

≍ r ∫_{r}^{∞} \frac{(ν × ℒ)(B((y, s), h))}{h^{3}} dh

≍ r ∫_{r}^{∞} \frac{ψ(ν × ℒ)(B(ψ(y, s), h))}{h^{3}} dh

≍ \frac{F_{2}^{ψ(ν×ℒ)} (ψ(y, s), r)}{r} = \frac{F μ̃ ; 2(ψ(y, s), r)}{r}

for all (y, s) ∈ Ĩ and r > 0. Equation (4.2) follows from this equality and the definition of the image measure. By the notation A ≍ B we mean that there exists a constant c > 0 such that A/c ≤ B ≤ cA. Above this constant does not depend on r.

Theorem 3.6 tells us that dim_{p} P_{t}ν = dim_{1} ν for ℒ^{1} -almost all t ∈ [0, 1], and so by Lemma 4.3 and equality (4.2), dim_{p}(Φ ∘ Π)μ̃ ≥ dim_{1} ν + 1 = dim_{2} μ̃. On the other hand, by Theorem 3.1 and Lemma 3.2

dim_{p}(Φ ∘ Π)μ̃ = dim_{2}(Φ ∘ Π)μ̃ ≤ dim_{2} μ̃

proving the claim.

Proof of Theorem 4.2. Theorem 4.2 follows directly from the previous lemma, since Φ does not change the dimension as a bi-Lipschitz mapping.

5. An example of a locally invariant measure whose packing dimension decreases under the projection

While the Hausdorff dimension of an invariant measure is preserved under the natural projection in the two-dimensional case, the packing dimension may change when the measure is projected to the base manifold. An example showing that the packing dimension of a locally invariant measure can really decrease under the projection can be obtained by using the measure of Example 5.1 in [FM]. In that example Falconer and Mattila constructed for every 0 < d < s < 2, d < 1 a finite, compactly supported measure ν_{s,d} on [0, 1]^{2} , whose packing dimension is s, and

dim_{1} ν_{s,d} = \frac{s(1 − \frac{1}{2}d)}{1 + \frac{1}{2}s − d} < s.

Using the similar notation as in the previous section we define

μ̃ := ψ(ν_{s,d} × ℒ^{1}),

500 Mika Leikas

in which case μ̃ is a locally invariant measure on SM and

dim_{p} Πμ̃ = dim_{2} μ̃ = dim_{1} ν_{s,d} + 1 < s + 1 = dim_{p} μ̃.

Remark 5.1. The example above suggests the existence of a globally invariant measure whose dimension decreases under the projection, but the construction of such a measure does not seem to be quite simple.

Acknowledgements. I would like to express my gratitude to my advisors Esa Järvenpää and Maarit Järvenpää for their help during this work. I would also like to thank Ville Suomala and Juhani Takkinen for many discussions and advice and help in using L^{A}TEX.

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Received 2 June 2004