Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 30, 2005, 237–260

AN EARTHQUAKE VERSION OF THE JACKSON–ZYGMUND THEOREM

F. P. Gardiner and J. Hu

Brooklyn College, CUNY, Department of Mathematics Brooklyn, New York 11210, U.S.A.; fgardiner@gc.cuny.edu, jun@sci.brooklyn.cuny.edu

Abstract. In this paper we present an analogy to the Jackson–Zygmund theorem on uniform approximability of Zygmund bounded continuous periodic functions by trigonometric polynomials. In our analogy the role of trigonometric polynomials is played by finite infinitesimal earthquakes.

1. Introduction

In his 1945 paper [10] Zygmund gave three proofs that the class Λ^{∗} of realvalued continuous periodic functions f with the property

(1)  |f(x) − \frac{f(x + t) + f(x − t)}{2}| = O(t)

is invariant under the Hilbert transform. One of the proofs employed the Jackson– Zygmund theorem which gives a criterion for a continuous periodic function to belong to the class Λ^{∗} in terms of the rate of its uniform approximability by trigonometric polynomials of a given degree. More precisely, assume f is continuous on R and f(x + 2π) = f(x). Let

(2)  ε_{f}(n) = inf_{T_{n}} ∥f − T_{n}∥_{∞},

where T_{n} is any trigonometric polynomial of degree less than or equal to n. That is, the infimum of the maximum difference between f(x) and T_{n}(x) is taken over all possible linear combinations T_{n} of the functions

1, cos x, sin x, cos 2x, sin 2x, …, cos nx, sin nx.

Jackson–Zygmund theorem [10]. A continuous periodic function f belongs to the class Λ^{∗} if and only if there is a constant C such that for all n ∈ N,

(3)  ε_{f}(n) ≤ C/n.

2000 Mathematics Subject Classification: Primary 37E10; Secondary 30F60, 37E20.

The necessity of this condition was shown by Jackson in [7] in 1912; he showed that the convolution of f with the Jackson kernel

K_{n}(x) = (\frac{sin(nx/2)}{sin(x/2)})^{4}

is a trigonometric polynomial T_{n} of degree 2n + 1 with the property that

∥f − T_{n}∥_{∞} ≤ C/n.

Zygmund showed in [10] that condition (3) was also sufficient by using a scaling argument and the inequality ∥P′′_{n}(x)∥_{∞} ≤ n^{2}∥P_{n}(x)∥_{∞} for any trigonometric polynomial P_{n}(x) of degree n. One can also find expository proofs of the Jackson– Zygmund theorem in [11] and [8, pp. 55–61].

In this paper we present an analogy to the Jackson–Zygmund theorem on uniform approximations of continuous periodic functions f ∈ Λ^{∗} by trigonometric polynomials. In our analogy the role of trigonometric polynomials is played by finite infinitesimal earthquakes, which are finite linear combinations of parabolic bump functions.

In order to state briefly our main result, we consider continuous periodic functions as continuous vector fields defined on the unit circle S^{1} = {z : |z| = 1}. Corresponding to such a real-valued, periodic function Ṽ with period 2π, the associated vector field V on S^{1} is given by the formula

(4)  V (e^{iθ}) = Ṽ(θ)ie^{iθ}.

V is said to be Zygmund bounded if

(5)  |V (e^{iθ}) − \frac{V (e^{i(θ+t)}) + V (e^{i(θ−t)})}{2}| ≤ Mt

for a constant M > 0 and for all 0 ≤ θ < 2π and 0 < t < π. Note that inequality (5) is equivalent to the existence of a constant M̃ such that

(6)  |Ṽ(θ) − \frac{Ṽ(θ + t) + Ṽ(θ − t)}{2}| ≤ M̃t,

and this condition coincides with condition (1) which defines the Zygmund class Λ^{∗}.

A vector field L(z) on S^{1} is said to be trivial if it is the initial tangent vector of a curve h_{ε} of Möbius transformations that preserves S^{1} with h_{0} equal to the identity map. Such a curve shares the same initial vector of a curve given by the formula

(7)  ε ↦ (z ↦ \frac{(1 + εA)z + εB}{ε\overline{B} z + (1 + εĀ)} ),

where A and B are arbitrary complex constants and z ∈ S^{1} . The tangent vector at ε = 0 to the curve (7) is

L(z) = (−\overline{B} z^{2} + (A − Ā)z + B).

If we transform L to L̃ by formula (4), that is, by the change of coordinate z ↦ θ where z = e^{iθ}, after a simplification we obtain

L̃(θ) = \frac{L(e^{iθ})}{ie^{iθ}} = a_{0} + a_{1} cos θ + b_{1} sin θ

for some real constants a_{0} , a_{1} and b_{1} .

Let S_{n} = {e^{2πik/2^{n}} : k = 0, 1, 2, …, 2^{n} − 1}, which is a set of 2^{n} equally spaced points on S^{1} . Consider the open unit disk D bounded by S^{1} as the hyperbolic plane. Let ℒ_{n} denote a collection of finitely many non-intersecting geodesics in D connecting points in S_{n} to points in S_{n} and σ_{n} be a measure supported on ℒ_{n} and assigning nonnegative weights to the geodesics in ℒ_{n}. For such a measure σ_{n} and a point z in S^{1} = ∂D, we define

(8) Vσ_{n}^{+}(z) = ∫∫ E_{ab}(z) dσ_{n}(a, b) and Vσ_{n}^{−}(z) = − ∫∫ E_{ab}(z) dσ_{n}(a, b),

where

(9)  E_{ab}(z) = \{ \begin{matrix}\frac{(z − a)(z − b)}{a − b} & for z ∈ [a, b], \\ 0 & otherwise. \end{matrix}

Here, for each geodesic line \overline{ab} in D with endpoints a and b on S^{1} , we denote by [a, b] the shorter arc on S^{1} with endpoints a and b. We label the endpoints so that going from a to points in the interval [a, b] to b is the counterclockwise direction. In the case that a and b are antipodal, we choose either of the two possible ways to label its endpoints by a and b and again we make going from a to points of [a, b] to b be the counterclockwise direction. Then Vσ_{n}^{+} and Vσ_{n}^{−} define Zygmund bounded vector fields on S^{1} . In fact, they are the initial tangent vectors to right and left earthquake curves (see [2]).

Let V be a continuous vector field on S^{1} and V_{n} = V |_{S_{n}} be the restriction of V on S_{n}. By the so-called finite infinitesimal earthquake theorem in [1], for each n ∈ N, there exists a measure σn^{+} supported on a finite lamination ℒn^{+} and a trivial vector field Ln^{+} such that

V_{n} = (Vσ_{n}^{+}^{+} + Ln^{+})|_{S_{n}}

and another measure σn^{−} supported on another finite lamination ℒn^{−} and a trivial vector field Ln^{−} such that

V_{n} = (Vσ_{n}^{−}^{−} + Ln^{−})|_{S_{n}}.

Define δ_{V} (n) by

(10)  δ_{V} (n) = ∥(Vσ_{n}^{+}^{+} + Ln^{+}) − (Vσ_{n}^{−}^{−} + Ln^{−})∥_{∞}.

In Sections 2 and 3, we prove the following main theorem.

Theorem 1. A continuous vector field V on S^{1} is Zygmund bounded if and only if there exists a constant C > 0 such that for all n ∈ N,

(11)  δ_{V} (n) ≤ C/2^{n},

where the constant C depends on the constant M which characterizes the Zygmund boundedness of V .

In the course of proving the necessity, we obtain

(12)  ∥V − (Vσ_{n}^{+}^{+} + Ln^{+})∥_{∞} ≤ C/2^{n}

and

(13)  ∥V − (Vσ_{n}^{−}^{−} + Ln^{−})∥_{∞} ≤ C/2^{n}.

Conversely, in Section 4 we give a counterexample to show that neither (12) nor (13) is sufficient by itself to imply that V is Zygmund bounded.

The supporting geodesics of σm^{+} (resp. σm^{−}) have no relation to the supporting geodesics of any other σn^{+} ’s (resp. σn^{−} ’s). However, we show in Section 5 that it is possible to approximate Zygmund bounded vector fields on S^{1} in the L_{∞} -norm by finite sums of parabolic bump functions in such a way that the supports of the measures σn^{+} are increasing sequences of nested sets and such that inequality (12) holds.

2. Proof of necessity

Let S_{n} = {e^{2πik/2^{n}} : k = 0, 1, 2, …, 2^{n} − 1}.

Lemma 1. If a Zygmund bounded vector field V on S^{1} vanishes on S_{n} for some n ∈ N, then

∥V ∥_{∞} ≤ C/2^{n}

for a constant C > 0 only depending on the Zygmund constant of V .

Proof. Denote by t_{n+k} = 1/2^{n+k}, k ∈ N.

Clearly, ∥V |_{S_{n}}∥_{∞} = 0. For any e^{2πix} ∈ S_{n+1} \ S_{n}, e^{2πi(x±t_{n+1})} ∈ S_{n} and then V (e^{2πi(x±t_{n+1})}) = 0. Hence

|V (e^{2πix})| = |V (e^{2πix}) − \frac{V (e^{2πi(x+t_{n+1})}) + V (e^{2πi(x−t_{n+1})})}{2}| ≤ Ct_{n+1}.

Therefore ∥V |_{S_{n+1}}∥_{∞} ≤ Ct_{n+1} .

Now for any e^{2πix} ∈ S_{n+2} \ S_{n+1} , e^{2πi(x±t_{n+2})} ∈ S_{n+1} . Then
|V (e^{2πix})| ≤ |V (e^{2πix}) − \frac{V (e^{2πi(x+t_{n+2})}) + V (e^{2πi(x−t_{n+2})})}{2}|

+ |\frac{V (e^{2πi(x+t_{n+2})}) + V (e^{2πi(x−t_{n+2})})}{2}|

≤ Ct_{n+2} + ∥V |_{S_{n+1}}∥_{∞} ≤ Ct_{n+2} + Ct_{n+1}.

Hence

∥V |_{S_{n+2}}∥_{∞} ≤ C(t_{n+1} + t_{n+2}).

Inductively, we obtain

∥V |_{S_{n+k}}∥_{∞} ≤ C ∑_{i=1}^{k} t_{n+i}.

Let S = ⋃_{n=1}^{∞} S_{n}. Then

∥V |_{S}∥_{∞} ≤ C ∑_{i=1}^{∞} t_{n+i} = \frac{C}{2^{n}} .

By the continuity of V and the density of S in S^{1} , we obtain

∥V ∥_{∞} ≤ \frac{C}{2^{n}} .

By a finite earthquake measure σ we mean a collection of nonnegative weights assigned to finitely many nonintersecting geodesics in the hyperbolic plane D. Let σ be such a measure and ℒ denote the collection of the finitely many geodesics which support σ. Let β be a closed hyperbolic geodesic segment in D with hyperbolic length l(β) less than or equal to 1. We denote by σ(β) the sum of the weights of the geodesics in ℒ intersecting β. In [9], Thurston defines the norm of σ to be

(14)  ∥σ∥_{Th} = sup_{l(β)≤1} σ(β) = sup_{l(β)=1} σ(β),

where β is a closed geodesic segment transversal to ℒ and l(β) denotes the hyperbolic length of β. As the same as defined in the introduction, we let

(15)  Vσ^{+}(x) = ∫∫ E_{ab}(x) dσ(a, b) and Vσ^{−}(x) = − ∫∫ E_{ab}(x) dσ(a, b).

Let V be a Zygmund bounded vector field on S^{1} . One can introduce a cross-ratio norm to measure the Zygmund bound of V . Given a quadruple Q = {a, b, c, d} consisting of four points a, b, c, d on the unit circle S^{1} arranged in counterclockwise order, we denote by

(16)  cr(Q) = \frac{(b − a)(d − c)}{(c − b)(d − a)}

and

(17)  V [Q] = \frac{V (b) − V (a)}{b − a} − \frac{V (c) − V (b)}{c − b} + \frac{V (d) − V (c)}{d − c} − \frac{V (d) − V (a)}{d − a}.

Then the cross-ratio norm ∥V ∥_{cr} is defined to be

(18)  ∥V ∥_{cr} = sup_{cr(Q)=1} |V [Q]|.

It is easy to see that ∥Vσ^{+}∥_{cr} = ∥Vσ^{−}∥_{cr} . Therefore, we simply use V_{σ} to denote either Vσ^{+} or Vσ^{−}.

By Theorem 4.1 in [1] or Theorem 4 in Section 18.4 in [3], ∥V_{σ}∥_{cr} ≤ C∥σ∥_{Th} for a constant C > 0. In [4] and [5], it is shown that the Thurston norm of σ is equivalent to the cross-ratio norm of V_{σ}.

Norm-equivalence theorem. There exists a universal constant C > 0 such that for any finite earthquake measure σ,

\frac{1}{C} ∥σ∥_{Th} ≤ ∥V_{σ}∥_{cr} ≤ C∥σ∥_{Th}.

Now we begin the proof of the necessity part of Theorem 1.

Proof. Let V be a Zygmund bounded vector field on S^{1} and V_{n} = V |_{S_{n}}. Let σn^{+} , σn^{−}, Ln^{+} and Ln^{−} be the same as defined in the introduction. By the same method in [5] used to show ∥σ∥_{Th} ≤ C∥V_{σ}∥_{cr} in the norm-equivalence theorem, one can show that there exists a universal constant C > 0 such that for all n,

∥σn^{+}∥_{Th} ≤ C∥V ∥_{cr} and ∥σn^{−}∥_{Th} ≤ C∥V ∥_{cr}.

Then the norm-equivalence theorem implies

∥Vσ_{n}^{+}^{+}∥_{cr} ≤ C∥σn^{+}∥_{Th} ≤ C′∥V ∥_{cr}.

Similarly, we have

∥Vσ_{n}^{−}^{−}∥_{cr} ≤ C′′∥V ∥_{cr}.

Therefore both Vσ_{n}^{+}^{+} and Vσ_{n}^{−}^{−}, and hence Vσ_{n}^{+}^{+} + Ln^{+} and Vσ_{n}^{−}^{−} + Ln^{−}, are Zygmund bounded with their bounds only depending on the Zygmund constant of V . The proof of necessity follows by applying Lemma 1 to the difference (Vσ_{n}^{+}^{+} + Ln^{+}) −

(Vσ_{n}^{−}^{−} + Ln^{−}).

3. Proof of sufficiency

Again let S_{n} = {e^{2πik/2^{n}} : k = 0, 1, 2, …, 2^{n} − 1}.

Proposition 1. A continuous vector field V on S^{1} is Zygmund bounded if and only if there exists a constant C > 0 such that for any n ∈ N and any triple {x − t, x, x + t} of three consecutive points in S_{n},

|V (x) − \frac{V (x − t) + V (x + t)}{2}| ≤ C|t|.

Proof. The necessity of the statement is obvious. To prove the sufficiency, we must show that if there exists a constant C > 0 such that for any n ∈ N and any triple {x − t, x, x + t} of three consecutive points in S_{n},

(19)  |V (x) − \frac{V (x − t) + V (x + t)}{2}| ≤ C|t|,

then V is Zygmund bounded.

Let S = ⋃_{n=1}^{∞} S_{n}. Since S is a dense subset of S^{1} , by the continuity of V it is sufficient to show for any symmetric triple {x − t, x, x + t} of three points in S, inequality (19) holds.

Let f(x) = V (e^{2πix})/ie^{2πix}. Then f is a periodic continuous function on R with period 1. Let B_{n} = {k/2^{n} : k = 0, 1, 2, …, 2^{n}} and B = ⋃_{n=1}^{∞} B_{n}. It is sufficient to show that for any symmetric triple {x − t, x, x + t} contained in B, f satisfies

(20)  |\frac{f(x + t) − f(x)}{t} − \frac{f(x) − f(x − t)}{t}| < C_{1}

for a constant C_{1} > 0.

Let n be the smallest integer such that the interval [x−t, x+t] contains three consecutive points in B_{n}, and denote them by x_{n} − t_{n}, x_{n} and x_{n} + t_{n}. Without loss of generality, we may assume x − t < x_{n} − t_{n} < x < x_{n} < x_{n} + t_{n} < x + t. Then there exist integers n < n_{1} < n_{2} < ⋯ < n_{l}, n < m_{1} < m_{2} < ⋯ m_{j} and n ≤ k_{1} < k_{2} < ⋯ < k_{i} such that

x + t = x_{n} + t_{n} + ∑_{s=1}^{l} \frac{1}{2^{n_{s}}} , x = x_{n} − ∑_{s=1}^{j} \frac{1}{2^{m_{s}}} and x − t = x_{n} − t_{n} − ∑_{s=1}^{i} \frac{1}{2^{k_{s}}} .

Let x_{n_{0}} = x_{n} + t_{n} and

x_{n_{s}} = x_{n} + t_{n} + \frac{1}{2^{n_{1}}} + \frac{1}{2^{n_{2}}} + ⋯ + \frac{1}{2^{n_{s}}}

for each 1 ≤ s ≤ l. Denote by

r_{R} = r_{n_{0}} = \frac{f(x_{n} + t_{n}) − f(x_{n})}{t_{n}} and r_{n_{s}} = \frac{f(x_{n_{s}}) − f(x_{n_{s−1}})}{x_{n_{s}} − x_{n_{s+1}}}

for each 1 ≤ s ≤ l.

244 F. P. Gardiner and J. Hu

Sublemma. Let x = −1/2^{m}, y = 0 and z = 1/2^{n} with m < n. Then

|\frac{f(z) − f(y)}{z − y} − \frac{f(y) − f(x)}{y − x}| ≤ (n − m + 2)C,

where C is the constant satisfying the inequality (19).

Proof. Let z_{k} = 2^{k}z for each 1 ≤ k ≤ n − m. Since y, z and z_{1} satisfy (19),

|f(y) + f(z_{1}) − 2f(z)| ≤ 2C(z − y).

Hence

|\frac{f(z_{1}) − f(y)}{2(z − y)} − \frac{f(z) − f(y)}{z − y}| ≤ C,

that is,

|\frac{f(z) − f(y)}{z − y} − \frac{f(z_{1}) − f(y)}{z_{1} − y}| ≤ C.

By the same reasoning, for each 1 ≤ k ≤ n − m − 1,

|\frac{f(z_{k}) − f(y)}{z_{k} − y} − \frac{f(z_{k+1}) − f(y)}{z_{k+1} − y}| ≤ C.

By the triangle inequality,

|\frac{f(z) − f(y)}{z − y} − \frac{f(z_{n−m}) − f(y)}{z_{n−m} − y}| ≤ (n − m)C.

Clearly,

|\frac{f(z_{n−m}) − f(y)}{z_{n−m} − y} − \frac{f(y) − f(x)}{y − x}| ≤ 2C.

The last two inequalities imply the sublemma.

Let C_{3} = 3C. By the previous sublemma, for each 1 ≤ s ≤ l,

|r_{n_{s}} − r_{n_{s−1}}| ≤ (n_{s} − n_{s−1} + 2)C ≤ 3(n_{s} − n_{s−1})C = C_{3}(n_{s} − n_{s−1}), and hence

|r_{n_{s}} − r_{R}| = |r_{n_{s}} − r_{n_{0}}| ≤ C_{3}(n_{s} − n_{0}) = C_{3}(n_{s} − n),

where n_{0} = n.

Similarly, let x_{m_{0}} = x_{n} and

x_{m_{s}} = x_{n} − (\frac{1}{2^{m_{1}}} + \frac{1}{2^{m_{2}}} + ⋯ + \frac{1}{2^{m_{s}}} )

An earthquake version of the Jackson–Zygmund theorem 245

for each 1 ≤ s ≤ j. Denote by

r′_{m_{0}} = r_{R} and r′_{m_{s}} = \frac{f(x_{m_{s−1}}) − f(x_{m_{s}})}{x_{m_{s−1}} − x_{m_{s}}}

for each 1 ≤ s ≤ j. By the same reasoning, we have for each 1 ≤ s ≤ j,

|r′_{m_{s}} − r′_{m_{s−1}}| ≤ C_{3}(m_{s} − m_{s−1}),

and hence

|r′_{m_{s}} − r_{R}| = |r′_{m_{s}} − r′_{m_{0}}| ≤ C_{3}(m_{s} − m_{0}) = C_{3}(m_{s} − n),

where m_{0} = n.

Clearly, x_{m_{j}} = x and x_{n_{l}} = x + t. Then

f(x + t) − f(x) = ∑_{s=1}^{j}[f(x_{m_{s−1}}) − f(x_{m_{s}})]

+ [f(x_{n} + t_{n}) − f(x_{n})] + ∑_{s=1}^{l}[f(x_{n_{s}}) − f(x_{n_{s−1}})]

= ∑_{s=1}^{j} r′_{m_{s}}(x_{m_{s−1}} − x_{m_{s}}) + r_{R}t_{n} + ∑_{s=1}^{l} r_{m_{s}}(x_{n_{s}} − x_{n_{s+1}})

= ∑_{s=1}^{j}(r′_{m_{s}} − r_{R})(x_{m_{s−1}} − x_{m_{s}})

+ ∑_{s=1}^{l}(r_{m_{s}} − r_{R})(x_{n_{s}} − x_{n_{s+1}}) + r_{R}t,

where

t = t_{n} + ∑_{s=1}^{j}(x_{m_{s−1}} − x_{m_{s}}) + ∑_{s=1}^{l}(x_{s} − x_{s+1}) = \frac{1}{2^{n}} + ∑_{s=1}^{j} \frac{1}{2^{m_{s}}} + ∑_{s=1}^{l} \frac{1}{2^{n_{s}}} .

Therefore

|\frac{f(x + t) − f(x)}{t} − r_{R}|

≤ \frac{∑_{s=1}^{j} |r′_{m_{s}} − r_{R}|(x_{m_{s−1}} − x_{m_{s}}) + ∑_{s=1}^{l} |r_{m_{s}} − r_{R}|(x_{s} − x_{s+1})}{t}

≤ C_{3} \frac{∑_{s=1}^{j} \frac{m_{s} − n}{2^{m_{s}}} + ∑_{s=1}^{l} \frac{n_{s} − n}{2^{n_{s}}}}{\frac{1}{2^{n}} + ∑_{s=1}^{j} \frac{1}{2^{m_{s}}} + ∑_{s=1}^{l} \frac{1}{2^{n_{s}}}}

= C_{3} \frac{∑_{s=1}^{j} \frac{m_{s} − n}{2^{m_{s}−n}} + ∑_{s=1}^{l} \frac{n_{s} − n}{2^{n_{s}−n}}}{1 + ∑_{s=1}^{j} \frac{1}{2^{m_{s}−n}} + ∑_{s=1}^{l} \frac{1}{2^{n_{s}−n}}} .

Since the series ∑_{n=1}^{∞} n/2^{n} converges, the numerator of the above quotient is bounded. Therefore there exists a constant C_{4} > 0 such that

|\frac{f(x + t) − f(x)}{t} − r_{R}| ≤ C_{4}.

Similarly, we can also show

|\frac{f(x) − f(x − t)}{t} − r_{L}| ≤ C_{5}

for a constant C_{5} > 0, where r_{L} = (f(x_{n}) − f(x_{n} − t_{n}))/(t_{n}). Since |r_{R} − r_{L}| ≤ C_{1} , we obtain

|\frac{f(x + t) − f(x)}{t} − \frac{f(x) − f(x − t)}{t}| ≤ C_{4} + C_{5} + C_{1},

which completes the proof.

Now we briefly recall the procedure in [1] for constructing the infinitesimal left earthquake V_{σ} associated to a vector field V defined on a finite subset A of S^{1} . For convenience, we work with the real line R and the upper half plane H instead of the circle S^{1} and the unit open disk D. Assume that A is a subset of R and V takes values only at the points in A. This procedure yields a finite lamination ℒ consisting of non-intersecting hyperbolic geodesics l in the upper half plane H together with nonnegative weights ϱ_{l} associated to each geodesic l in ℒ .

To describe this procedure we refer to the example illustrated in Figure 1, which treats a case where A∪{∞} consists of 9 points. We will need the formulae for the parabolic bump function E_{ab}(x) given similarly in (9) as

(21)  E_{ab}(x) = \{ \begin{matrix}\frac{(x − a)(x − b)}{a − b} & for a ≤ x ≤ b, \\ 0 & otherwise \end{matrix}

and the special cases

(22)  E_{a∞}(x) = \{ \begin{matrix}x − a & for a ≤ x ≤ ∞, \\ 0 & otherwise \end{matrix}

and

(23)  E_{−∞b}(x) = \{ \begin{matrix}−(x − b) & for −∞ ≤ x ≤ b, \\ 0 & otherwise. \end{matrix}

Figure 1.

The procedure consists of the following three steps.

Step 1. Draw a line l_{1} which meets the graph of V over A at the point furthest to the left, which meets the graph of V at at least one more point p and which lies on or below the graph. In Figure 1, this is the line l_{1} and p = x_{4} .

Step 2. If p is the furthest right point on the graph of V on A, then stop. Otherwise draw a line l_{2} which meets the graph of V at the point (p, V (p)), meets the graph of V at at least one more point q and which lies on or below the graph of V . Replace p by q, continue this step inductively until p is the furthest right point on the graph of V on A. Then we obtain a finite sequence of the line segments l_{1}, l_{2}, …, l_{k}. In Figure 1 there are three such line segments, l_{1} , l_{2} and l_{3} .

Step 3. Let R(x) be the piecewise linear function whose graph consists of these line segments. Over each linear piece of the graph of R(x) add a function ϱ_{ab}E_{ab}(x) where a and b are the left and right endpoints of the linear piece, and ϱ_{ab} ≥ 0 and is as large as possible so that the graph of R(x) + ϱ_{ab}E_{ab}(x) lies on or below the graph of V .

Continue inductively to add parabolic bump functions ϱ_{a_{j}b_{j}}E_{a_{j}b_{j}}(x) in the above way until the graph of the resulting function passes through all points on the graph of V . Now we obtain a function Ṽ(x) of the form

(24)  Ṽ(x) = ∑_{a_{j}b_{j}} ϱ_{a_{j}b_{j}}E_{a_{j}b_{j}}(x) + B(x)

where (a_{j}, b_{j})’s are pairs of points in A ∪ {∞}, the hyperbolic goedesics l_{j} with endpoints a_{j} and b_{j} do not intersect, and B(x) is affine. Moreover, it satisfies

(i) Ṽ(x_{j}) = V (x_{j}) for each x_{j} in A and

(ii) the graph of each parabolic or linear segment of Ṽ, if extended, lies entirely on or below the graph of V .

In the example depicted in Figure 1, the weight assigned to the line connecting x_{4} to ∞ is equal to the slope of l_{2} minus the slope of l_{1} and the weight assigned to the line connecting x_{7} to ∞ is equal to the slope of l_{3} minus the slope of l_{2} . All weights are given by the numbers ϱ_{a_{j}b_{j}} in the sum (24). For example, the weight assigned to the geodesic connecting x_{4} to x_{7} is the coefficient of E_{x_{4}x_{7}} in the sum (24).

Figure 2 is the translation of Figure 1 from the upper half plane to the unit disk.

Figure 2.

We obtain the following proposition.

Proposition 2. Assume that f is a function defined on a set A consisting of finitely many points on the real line R. Suppose that E is the unique finite infinitesimal left (resp. right) earthquake on H which coincides with f on A. For any two adjacent points X and Y on the graph of f, let E′ be the collection of the summands of E which contribute values at X and/or Y , and Q be the sum of the summands in E′. Then the graph of Q passes through X and Y , and all other points on the graph of f are not below (resp. above) the graph of Q.

In order to prove the sufficiency of the condition in Theorem 1, we also need the following elementary lemmas and corollaries.

Lemma 2. Given t > 0 and h > 0, let X = (−t, 0), Y = (0, h) and Z = (t, 0). Suppose that f(x) = ax^{2} + bx + c is a quadratic polynomial function with a < 0 such that X and Y are on the graph of f and Z is not below the graph. Let L(x) be the piecewise linear function obtained by connecting X to Y and Y to Z. Then

(f − L)(−\frac{1}{2}t) ≥ \frac{1}{4}h.

Proof. Clearly, L(x) = (h/t)x + h for x ≤ 0 and

f(x) = ax(x + t) + \frac{h}{t}x + h.

Since f(t) = 2at^{2} + 2h ≤ 0, −at^{2} ≥ h. Then

(f − L)(\frac{1}{2}(−t)) = −\frac{1}{2}at(\frac{1}{2}(−t) + t) = −\frac{1}{4}at^{2} ≥ \frac{1}{4}h.

Corollary 1. Suppose that t > 0 and y − \frac{1}{2}(x + z) = h > 0, and let X = (−t, x), Y = (0, y) and Z = (t, z). Under the same notation and assumptions as in the previous lemma 2, we also have

(f − L)(−\frac{1}{2}t) ≥ \frac{1}{4}h.

Similarly, we obtain

Lemma 3. Given t > 0 and h < 0, let X = (−t, 0), Y = (0, h) and Z = (t, 0). Suppose that f(x) = ax^{2} + bx + c is a quadratic polynomial function with a > 0 such that X and Y are on the graph of f and Z is not above the graph. Let L(x) be the piecewise linear function obtained by connecting X to Y and Y to Z. Then

(L − f)(−\frac{1}{2}t) ≥ −\frac{1}{4}h.

Corollary 2. Suppose that t > 0 and y − \frac{1}{2}(x + z) = h < 0, and let X = (−t, x), Y = (0, y) and Z = (t, z). Under the same notation and assumptions in the previous lemma 3, we also have

(L − f)(−\frac{1}{2}t) ≥ −\frac{1}{4}h.

To show the sufficiency of the condition in Theorem 1, we proceed by making a contradiction.

Proof. Let V be a continuous vector field on S^{1} satisfying the condition (11) in our Theorem 1. Suppose that V is not Zygmund bounded. By Proposition 1, there exists an infinite sequence {T_{n}}_{n=1}^{∞} of symmetric triples T_{n−k} = {x_{n} − t_{n}, x_{n}, x_{n} + t_{n}} in S such that t_{n} → 0 and

|(V (x_{n}) − \frac{V (x_{n} − t_{n}) + V (x_{n} + t_{n})}{2}) / t_{n}| → +∞

as n → ∞. Therefore, by passing to a subsequence if necessary, we may assume

C_{n} = (V (x_{n}) − \frac{V (x_{n} − t_{n}) + V (x_{n} + t_{n})}{2}) / t_{n} → +∞

as n → ∞. We may also assume that C_{n} > 0 for all n.

For the convenience in estimates, we first convert the vector field V to the vector fields Ṽ_{n} on the real line R as follows.

Let B_{n}(z) = x_{n}(i − z)/(i + z). Then B_{n} is an orientation-preserving Möbius transformation mapping H onto D, i to the center of D, the origin of H to x_{n} and |B′_{n}(0)| = 2. Denote by

(25)  Ṽ_{n}(x) = \frac{V (B_{n}(x))}{B′_{n}(x)}|_{B_{n}^{−1}(S_{n})}.

Now let σ̃n^{+} = (B_{n}^{−1})^{∗}σn^{+} (resp. σ̃n^{−}) be the pushforward of σn^{+} (resp. σn^{−}) by B_{n}^{−1} . Denote by

(26) Vσ̃_{n}^{+}(x) = ∫∫ E_{ab}(x) dσ̃n^{+}(a, b) and Vσ̃_{n}^{−}(x) = − ∫∫ E_{ab}(x) dσ̃n^{−}(a, b),

where E_{ab} is defined by (21). It is proved in Lemma 1 in [5] that

(27)  Vσ̃_{n}^{+}(x) = \frac{Vσ_{n}^{+}(B_{n}(x))}{B′_{n}(x)}.

Similarly,

(28)  Vσ̃_{n}^{−}(x) = \frac{Vσ_{n}^{−}(B_{n}(x))}{B′_{n}(x)}.

Corresponding to a trivial vector field on S^{1} , a trivial vector field on R is just an affine map. Then by the finite infinitesimal earthquake theorem in [1] (see the introduction), there are affine maps L̃n^{+} and L̃n^{−} such that

Ṽ_{n} = (Vσ̃_{n}^{+} + L̃n^{+})|_{B_{n}^{−1}(S_{n})} = (Vσ̃_{n}^{−} + L̃n^{−})|_{B_{n}^{−1}(S_{n})}.

Clearly, B_{n}^{−1} maps a symmetric triple T_{n} to a triple T̃_{n} = {−t̃_{n}, 0, t̃_{n}}. Since |(B_{n}^{−1})′(x_{n})| = 1/|B′_{n}(0)| = \frac{1}{2} , all (B_{n}^{−1})’s are almost linear maps in small neighborhoods of x_{n} with the same asymptotic rescaling. Therefore as n → +∞,

C̃_{n} = [Ṽ_{n}(0) − \frac{Ṽ_{n}(−t̃_{n}) + Ṽ_{n}(t̃_{n})}{2}] / t̃_{n} → +∞.

Let X_{n} = (−t̃_{n}, Ṽ_{n}(−t̃_{n})), Y_{n} = (0, Ṽ_{n}(0)) and Z_{n} = (t̃_{n}, Ṽ_{n}(t̃_{n})). Let E_{n} be the collection of all quadratic polynomials comprising Vσ̃_{n}^{+} and E′_{n} the collection of the quadratic polynomials in E_{n} which contribute values at X_{n} and/or Y_{n}. Now let Qn^{+} be the sum of all polynomials in E′_{n}, then by Proposition 2, Z_{n} is not below the graph of Qn^{+} . Assume that L_{n} is the linear approximation of Vσ̃_{n}^{+} on T̃_{n}. By Corollary 1,

(Qn^{+} − L_{n})(−\frac{1}{2} t̃_{n}) ≥ \frac{1}{4} C̃_{n}t̃_{n} > 0.

On the interval [−t̃_{n}, 0], Q^{+} = Vσ̃_{n}^{+} and then

[(Vσ̃_{n}^{+} + L̃n^{+}) − (L_{n} + L̃n^{+})](−\frac{1}{2} t̃_{n}) = (Qn^{+} − L_{n})(−\frac{1}{2} t̃_{n}) ≥ \frac{1}{4} C̃_{n}t̃_{n}.

On the other hand, the graph of the right finite infinitesimal earthquake Vσ̃_{n}^{−} + L̃n^{−} is not above the graph of L_{n} + L̃n^{+} on the interval [−t̃_{n}, 0] since L_{n} + L̃n^{+} is equal to the linear approximation of Ṽ_{n} on this interval. Hence

[(L_{n} + L̃n^{+}) − (Vσ̃_{n}^{−} + L̃n^{−})](−\frac{1}{2} t̃_{n}) ≥ 0.

Therefore

[(Vσ̃_{n}^{+} + L̃n^{+}) − (Vσ̃_{n}^{−} + L̃n^{−})](−\frac{1}{2} t̃_{n}) ≥ \frac{1}{4} C̃_{n}t̃_{n}.

Since all B_{n} ’s are almost linear maps in small neighborhoods of the origin with the same asymptotic rescaling,

lim_{n→∞} \frac{t̃_{n}}{t_{n}} → 2.

By (27) and (28),

[(Vσ_{n}^{+}+Ln^{+})−Vσ_{n}^{−}+Ln^{−})](B_{n}(−\frac{1}{2} t̃_{n})) = B′_{n}(−\frac{1}{2} t̃_{n})[(Vσ̃_{n}^{+}+L̃n^{+})−(Vσ̃_{n}^{−}+L̃n^{−})](−\frac{1}{2} t̃_{n}).

As n → ∞,

|B′_{n}(−\frac{1}{2} t̃_{n})| → 2 and C̃_{n} → +∞,

and therefore

\frac{∥(Vσ_{n}^{+} + Ln^{+}) − Vσ_{n}^{−} + Ln^{−})∥_{∞}}{t_{n}} ≥ \frac{1}{4}|B′_{n}(−\frac{t̃_{n}}{2} )|\frac{t̃_{n}}{t_{n}} C̃_{n} → +∞,

which contradicts with the condition (11). This completes the proof of sufficiency.

4. Counterexamples

In this section we first give an example of a Zygmund unbounded vector field V on S^{1} which can be well approximated by its linear approximations on binary points. Then we show that neither the condition (12) nor the condition (13) on the approximations of V is sufficient by itself to imply that V is Zygmund bounded. For simplicity, we will construct such examples of V on the real line.

Proposition 3. There exists a Zygmund unbounded continuous function on R which can be approximated as well as in (12) by its piecewise linear approximations on binary points.

Proof. For each n ∈ N, define

(29)  f_{n}(x) = \{ \begin{matrix}\frac{1}{2^{n}} − |x − \frac{1}{2^{n}} | & for |x − \frac{1}{2^{n}} | ≤ \frac{1}{2^{n}}, \\ 0 & otherwise. \end{matrix}

Let V = ∑_{n=1}^{∞} f_{n}, S_{n} = {k/2^{n} : k ∈ Z} and S = ⋃_{n=0}^{∞} S_{n}. Suppose that V_{n} is the piecewise linear approximation of V on S_{n}. Then

∥V − V_{n}∥_{∞} ≤ ∑_{k=n+1}^{∞} \frac{1}{2^{k}} = \frac{1}{2^{n}} .

This means that V can be approximated by V_{n} ’s as well as in (12). On the other hand,

\frac{|V (0) − \frac{1}{2}(V (−t_{n}) + V (t_{n}))|}{t_{n}} = \frac{V (t_{n})}{t_{n}} = n → ∞

as n → ∞, where t_{n} = 1/2^{n}. Hence V is not Zygmund bounded.

Proposition 4. There is a Zygmund unbounded continuous function V on R and an infinite sequence {(σ_{n}, ℒ_{n})}_{n=1}^{∞} of earthquake measures σ_{n} supported on the laminations ℒ_{n} consisting of finitely many geodesics such that

(i) for each n and each geodesic \overline{ab} ∈ ℒ_{n}, |a − b| ≥ 1/2^{n};

(ii) for each n, ℒ_{n} ⊂ ℒ_{n+1} ; and

(iii) ∥V − V_{σ_{n}}∥_{∞} ≤ C/2^{n} for a constant C > 0 independent of n.

Proof. Let ℒ be the lamination consisting of all geodesics l_{n} connecting the origin to 1/2^{n} for all n ∈ {0} ∪ N and assume the earthquake measure σ has the weight 1 on each geodesic in ℒ .

For each n ∈ N, let ℒ_{n} be the lamination consisting of all geodesics \overline{ab} ∈ ℒ with |a − b| ≥ 1/2^{n} and σ_{n} be the restriction of σ on ℒ_{n}.

Let V = Vσ^{+} and V_{n} = Vσ_{n}^{+} . By the norm-equivalence theorem, V_{n} is a Zygmund bounded vector field on R for each n ∈ N since ∥σ_{n}∥_{Th} < n. It is easy to see that V (x) is continuous at any x ≠ 0; and the continuity of V (x) at x = 0 is implied by the following that

0 < V (\frac{1}{2^{n}} ) ≤ \frac{n}{2^{n}} → 0 as n → ∞.

Now we estimate ∥V − V_{n}∥_{∞}. Clearly, for each k ∈ {0} ∪ N, ∥V_{σ|_{l_{k}}} ∥_{∞} = \frac{1}{4}(1/2^{k}). Then

|V (x) − V_{n}(x)| = | ∑_{k=n+1}^{∞} V_{σ|_{l_{k}}} (x)| ≤ ∑_{k=n+1}^{∞} ∥V_{σ|_{l_{k}}} ∥_{∞} = \frac{1}{4} ∑_{k=n+1}^{∞} \frac{1}{2^{k}} = \frac{1}{4} \frac{1}{2^{n}} .

Hence

(30)  ∥V − V_{n}∥_{∞} ≤ \frac{C}{2^{n}}

for a constant C > 0 independent of n.

It remains to check that V is not Zygmund bounded. This is true because of the norm-equivalence theorem for general earthquakes in the next section, or we can prove it as follows.

By the norm-equivalence theorem for finite earthquakes, ∥V_{n}∥_{cr} → ∞ since ∥σ_{n}∥_{Th} = n → ∞ as n → ∞. From (30), we can show ∥V ∥_{cr} = ∞.

The same example constructed in the proof of the previous proposition implies

Proposition 5. There exists a Zygmund unbounded continuous function V on R and an infinite sequence {(σ_{n}, ℒ_{n})}_{n=1}^{∞} of earthquake measures supported on the laminations ℒ_{n} consisting of finitely many geodesics such that

(1) for each n and each geodesic \overline{ab} ∈ ℒ_{n}, |a − b| ≥ 1/n;

(2) for each n, ℒ_{n} ⊂ ℒ_{n+1} ; and

(3) ∥V − V_{σ_{n}}∥_{∞} ≤ C/n for a constant C > 0 independent of n.

Proof. We use the same notation in the proof of the previous proposition except we let ℒ_{n} be the lamination consisting of all geodesics \overline{ab} ∈ ℒ with |a − b| ≥ 1/n and σ_{n} be the restriction of σ on ℒ_{n}. We only need to check

∥V − V_{n}∥_{∞} ≤ \frac{C}{n}

for a constant C > 0 independent of n.

For each n ∈ N, there exists m ∈ N such that 2^{m−1} ≤ n < 2^{m}. Then

|V (x) − V_{n}(x)| = | ∑_{k=m}^{∞} V_{σ|_{l_{k}}} (x)| ≤ ∑_{k=m}^{∞} ∥V_{σ|_{l_{k}}} ∥_{∞} = \frac{1}{4} ∑_{k=m}^{∞} \frac{1}{2^{k}} = \frac{1}{2^{m+1}} < \frac{1}{2} \frac{1}{n}.

This implies the previous inequality.

5. Nested laminations

In our main theorem the supporting geodesics of σm^{+} (resp. σm^{−}) have no relation with the supporting geodesics of any other σn^{+} (resp. σn^{−}). In this section we introduce a method to approximate uniformly Zygmund bounded vector fields on S^{1} by finite sums of parabolic bump functions in such a way that the supports of the measures are increasing sequences of nested sets.

A geodesic lamination ℒ in the hyperbolic plane D is a collection of geodesics which foliate a closed subset L of D. Here L is called the locus of ℒ , the geodesics are called the leaves of ℒ , the connected components of D\L are called the gaps, and the gaps and the leaves of ℒ are called the strata of ℒ .

Let S^{1} denote the boundary circle of D, and X the space S^{1} × S^{1} \ {the diagonal} factorized by the equivalence relation (a, b) ∼ (b, a). A Borel measure σ defined on X is called an earthquake measure if there is a lamination ℒ such that σ is supported on the pairs of the endpoints of the leaves in ℒ .

The Thurston norm of an earthquake measure σ can be defined as the same as the Thurston norm of finite earthquake measures in Section 2. More precisely, for any closed hyperbolic geodesic segment β in D of hyperbolic length ≤ 1, let l_{1} and l_{2} be the two geodesics which bound all geodesics in ℒ intersecting β and suppose the stripe S bounded by l_{1} and l_{2} in the unit disk is of the form [a, b] × [c, d], where a, d and b, c are the endpoints of l_{1} and l_{2} , respectively, and a, b, c, d are arranged on S^{1} in the counterclockwise order. Denote by

σ(β) = σ([a, b] × [c, d]).

The Thurston norm of σ is defined to be

(31)  ∥σ∥_{Th} = sup_{l(β)≤1} σ(β) = sup_{l(β)=1} σ(β),

where β is a closed geodesic segment transversal to ℒ and l(β) denotes the hyperbolic length of β. Let ℳ be the collection of all earthquake measures defined on X. Any measure σ ∈ ℳ is called Thurston bounded if it has finite Thurston norm.

Given σ ∈ ℳ , also denote by

(32)  V_{σ}(x) = E(σ)(x) = ∫∫ E_{ab}(x) dσ(a, b),

where E_{ab}(x) = ((x − a)(x − b))/(a − b) if x ∈ [a, b], and otherwise E_{ab}(x) = 0. Then V_{σ} defines a tangent vector field on S^{1} . The following two theorems hold in general.

Infinitesimal earthquake theorem ([1]). If σ is Thurston bounded then V_{σ} is Zygmund bounded; conversely, for any Zygmund bounded tangent vector field V on S^{1} , there exists a Thurston bounded earthquake measure σ such that

(33)  V (x) = π ∫∫ E_{ab}(x) dσ(a, b) modulo a quadratic polynomial;

and furthermore, if two V ’s differ by a quadratic polynomial then the corresponding σ ’s are the same.

Norm-equivalence theorem ([4] or [5]). There exists a universal constant C > 0 such that for any earthquake measure σ,

\frac{1}{C} ∥σ∥_{Th} ≤ ∥V_{σ}∥_{cr} ≤ C∥σ∥_{Th}.

Remark. There is an analogue of the norm-equivalence theorem for earthquake measures and the boundary homeomorphisms of the corresponding earthquake maps. This work was initiated in [2] and completed in [6].

In what follows, we show the following theorem.

Theorem 2. Let σ be a Thurston bounded earthquake measure. Then there exists an infinite sequence {(σ_{n}, ℒ_{n})}_{n=1}^{∞} of earthquake measures satisfying:

(i) σ_{n} ’s are uniformly Thurston bounded;

(ii) for each n, ℒ_{n} consists of finitely many geodesics in ℒ and for each geodesic \overline{ab} in ℒ_{n}, |a − b| ≥ 1/n, where a and b denote the endpoints of \overline{ab};

(iii) ℒ_{1} ⊂ ℒ_{2} ⊂ ℒ_{3} ⊂ ⋯ ⊂ ℒ_{n} ⊂ ⋯ ⊂ ℒ ; and

(iv) there exists a constant C > 0 such that for all n,

∥V_{σ} − V_{σ_{n}}∥_{∞} ≤ \frac{C}{n} .

Remark. The subsequence {σ_{2^{k}}, ℒ_{2^{k}}}_{k=1}^{∞} of {σ_{n}, ℒ_{n}}_{n=1}^{∞} in the previous theorem satisfies

∥V_{σ} − V_{σ_{2^{k}}} ∥_{∞} ≤ \frac{C}{2^{k}}

for any integer k ≥ 1.

We divide the proof into two steps. In the first step, let ℒ′ be the lamination obtained from ℒ by deleting the short geodesics (in the Euclidean metric) and σ′ be the restriction of σ on ℒ′, we show that V_{σ} differs from V_{σ′} by an amount commensurable to the maximal Euclidean length of the deleted geodesics. In the second step, we show that σ′ can be approximated by measures σ′′ supported on finitely many geodesics such that V_{σ′} differs from V_{σ′′} by an error as small as required.

Figure 3. The lamination \overline{ℛ} _{n}.

For each n ∈ N, let ℛ_{n} be the collection of the geodesics \overline{ab} in ℒ with |a−b| < 1/n and \overline{ℛ} _{n} be the collection of the geodesics \overline{ab} in ℒ with |a−b| ≤ 1/n. For each n, \overline{ℛ} _{n} can be written as a disjoint union of subcollections \overline{ℛ} _{n}^{α(n)} of \overline{ℛ} _{n} satisfying that for each \overline{ℛ} _{n}^{α(n)} there exists a geodesic \overline{ab} ∈ \overline{ℛ} _{n}^{α(n)} such that any geodesic in \overline{ℛ} _{n} connecting a point in [a, b] to a point in [a, b] belongs to \overline{ℛ} _{n}^{α(n)} , where [a, b] denote the shorter arc on S^{1} bounded by the endpoints of \overline{ab}. Assume ℛ_{n}^{α(n)} =\overline{ℛ} _{n}^{α(n)} ∩ ℛ_{n}. Clearly,

\overline{ℛ} _{n} = ⋃_{α(n)}\overline{ℛ} _{n}^{α(n)} and ℛ_{n} = ⋃_{α(n)} ℛ_{n}^{α(n)}.

Let σ̄^{α(n)} be the restriction of σ on \overline{ℛ} _{n}^{α(n)} and σ^{α(n)} be the restriction of σ on ℛ_{n}^{α(n)} . Let V_{σ^{α(n)}} = E(σ^{α(n)}).

Lemma 4. There exists a constant C > 0 only depending on ∥σ∥_{Th} such that

∥V_{σ^{α(n)}}∥_{∞} ≤ \frac{C}{n} .

Proof. Let V_{α(n)} = V_{}_{σ̄^{α(n)}} = E(σ̄^{α(n)}) and \overline{ab} be the unique geodesic in \overline{ℛ} _{n}^{α(n)}

such that any geodesic in \overline{ℛ} _{n}^{α(n)} connecting a point in [a, b] to a point in [a, b]. Clearly,

∥V_{σ^{α(n)}}∥_{∞} ≤ ∥V_{}_{σ̄^{α(n)}}∥_{∞} = ∥V_{α(n)}∥_{∞}.

Therefore we only need to show

∥V_{α(n)}∥_{∞} ≤ \frac{C}{n}

for a constant C > 0.

For any x ∉ (a, b), V_{α(n)}(x) = 0. For any x ∈ (a, b), let y be the point on S^{1} such that y, a, x, b are on S^{1} in the counter-clockwise order and cr({y, a, x, b}) = 1. Let Q = {y, a, x, b}. Since V_{α(n)}(y) = V_{α(n)}(a) = V_{α(n)}(b) = 0,

V_{α(n)}[Q] = −\frac{V_{α(n)}(x)}{x − a} − \frac{V_{α(n)}(x)}{b − x} = −V_{α(n)}(x)\frac{b − a}{(x − a)(b − x)}.

Hence

|V_{α(n)}(x)| = |V_{α(n)}[Q]|\frac{(x − a)(b − x)}{b − a} ≤ ∥V_{α(n)}∥_{cr} \frac{b − a}{4}.

Since ∥σ^{α(n)}∥_{Th} ≤ ∥σ∥_{Th} , by the norm-equivalence theorem,

∥V_{α(n)}∥_{∞} ≤ ∥V_{α(n)}∥_{cr} \frac{b − a}{4} ≤ C∥σ∥_{Th} \frac{b − a}{4} ≤ \frac{C∥σ∥_{Th}}{4} \frac{1}{n}.

It completes the proof.

Now let ℒ̃_{n} = ℒ \ℛ_{n}, that is the collection of the geodesics l = \overline{ab} in ℒ with |a − b| ≥ 1/n, and let σ̃_{n} be the restriction of σ on ℒ̃_{n}. Assume Vσ̃_{n} = E(σ̃_{n}).

Proposition 6. We obtain the inequality

∥V_{σ} − Vσ̃_{n}∥_{∞} ≤ \frac{C}{n} .

Proof. For each point x ∈ S^{1} and each n ∈ N, there exists α(n) such that

V_{σ}(x) − Vσ̃_{n}(x) = V_{σ^{α(n)}}(x).

By the previous lemma,

|V_{σ^{α(n)}}(x)| ≤ ∥V_{σ^{α(n)}}∥_{∞} ≤ \frac{C}{n} .

It implies the proposition.

Now we start to do the second step.

Given a lamination ℒ , two geodesics l_{1} and l_{2} in ℒ are said to be simply parallel to each other with respect to ℒ if any other geodesic l of ℒ contained in the stripe between l_{1} and l_{2} separates l_{1} and l_{2} . Simply parallel geodesics have the reflexive, symmetric and transitive properties. Given each geodesic l in ℒ , there is a unique maximal collection ℒ_{l} of the geodesics which are simply parallel to l with respect to ℒ . Since the locus of ℒ is a closed subset in the hyperbolic plane, the locus of ℒ_{l} is also so. Maximal collections of simply parallel geodesics with respect to a lamination are pairwise disjoint. Laminations without short geodesics in Euclidean metric have the following property.

Figure 4. The lamination ℒ_{n}.

Lemma 5. With respect to each lamination ℒ̃_{n}, there are only finitely many maximal collections of simply parallel geodesics.

Proof. For each n, the lamination ℒ̃_{n} has no geodesic \overline{ab} with |a−b| < 1/n. Arguing by contradiction, one can show that ℒ̃_{n} has only finitely many maximal collections of simply parallel geodesics.

One easily obtains the following two elementary lemmas.

Lemma 6. Let a < a′ < b′ < b be four points on the real line. Then

∥E_{ab} − E_{a′b′}∥_{∞} → 0

as a → a′ and b′ → b.

Lemma 7. Let ℒ be a collection of simply parallel geodesics in the upper half plane and σ a measure supported on ℒ . Suppose that a < a′ < b′ < b and all geodesics of ℒ are contained in the strip between the geodesics connecting a to b and a′ to b′ respectively. Then for each x on R,

σ(ℒ)E_{a′b′}(x) ≤ V_{σ}(x) ≤ σ(ℒ)E_{ab}(x)

and then

∥V_{σ} − σ(ℒ)E_{ab}∥_{∞} ≤ σ(ℒ)∥E_{ab} − E_{a′b′}∥_{∞}.

Lemmas 5, 6 and 7 imply

Proposition 7. For each n and any ε_{n} > 0, there exists an earthquake measure σ_{n} supported on a finite sublamination ℒ_{n} of ℒ̃_{n} such that ∥σ_{n}∥_{Th} ≤ 2∥σ_{n}∥_{Th} and

∥Vσ̃_{n} − V_{σ_{n}}∥_{∞} < ε_{n}.

Proof. For each n, let ℒ̃ _{n}^{i}, i = 1, 2, …, k(n), be the maximal collections of simply parallel goedesics with respect to ℒ̃_{n}. Let σ̃_{n}^{i} be the restriction of σ̃_{n} on ℒ̃ _{n}^{i} for each 1 ≤ i ≤ k(n). Then

Vσ̃_{n} = ∑_{i=1}^{k(n)} V_{}_{σ̃_{n}^{i}}.

For each i, we construct a finite earthquake measure (ℒ_{n}^{i}, σ_{n}^{i}) as follows. Let l_{a} and l_{b} be the two geodesics in ℒ̃ _{n}^{i} such that the strip between them contains all geodesics in ℒ̃ _{n}^{i}. Without loss of generality, we may assume l_{a} ≠ l_{b}. Let ℒ_{n}^{i} = {l_{1} = l_{a}, l_{2}, l_{2}, …, l_{m} = l_{b}} be a collection of finitely many geodesics contained in ℒ̃ _{n}^{i} in order. We construct σ_{n}^{i} by defining σ_{n}^{i}(l_{k}) to be σ̃_{n}^{i}(l_{k}) plus the measure of the geodesics of ℒ̃ _{n}^{i} contained in the open strip between l_{k} and l_{k+1} for 1 ≤ k ≤ m − 1 and σ_{n}^{i}(l_{m}) = σ̃_{n}^{i}(l_{m}). Since the total σ̃_{n}^{i} -measure of ℒ̃ _{n}^{i} is finite, by Lemmas 6 and 7, for an arbitrary small ε(i) > 0, there exists a finite earthquake measure (ℒ_{n}^{i}, σ_{n}^{i}) constructed as the same as the above such that

∥V_{}_{σ̃_{n}^{i}} − V_{σ_{n}^{i}}∥_{∞} < ε(i).

Let ℒ_{n} = ⋃_{i=1}^{k(n)} ℒ_{n}^{i} and σ_{n} = ∑_{i=1}^{k(n)} σ_{n}^{i}. It is easy to see that ∥σ_{n}∥_{Th} ≤ 2∥σ̃_{n}∥_{Th} . Given any ε_{n} > 0, let ε(i) < ε_{n}/k(n). Then

∥Vσ̃_{n} − V_{σ_{n}}∥_{∞} = ∑_{i=1}^{k(n)} ∥V_{}_{σ̃_{n}^{i}} − V_{σ_{n}^{i}}∥_{∞} < ε_{n}.

Furthermore, we obtain

Proposition 8. There exists an infinite sequence {(σ_{n}, ℒ_{n})}_{n=1}^{∞} of earthquake measures such that for each n ∈ N,

(i) ℒ_{n} is a finite subset of ℒ̃_{n} and ∥σ_{n}∥_{Th} ≤ 2∥σ̃_{n}∥_{Th} ,

(ii) ℒ_{n} ⊂ ℒ_{n+1} , and

(iii) ∥Vσ̃_{n} − V_{σ_{n}}∥_{∞} ≤ 1/n.

Proof. The condition (i) is obviously satisfied, and the previous proposition implies the condition (iii) by letting ε_{n} = 1/n. It remains to show how one can construct ℒ_{n+1} from ℒ_{n} such that ℒ_{n} ⊂ ℒ_{n+1} .

We divide the maximal collections ℒ̃ _{n+1}^{j} of simply parallel geodesics with

respect to ℒ̃_{n+1} into two groups. The group I consists of the collections ℒ̃ _{n+1}^{j(I)}

which overlaps maximal collections ℒ̃ _{n}^{i} of simply parallel geodesics with respect to

ℒ̃_{n}, and the group II consists of the collections ℒ̃ _{n+1}^{j(II)} which do not overlap any

maximal collection ℒ̃ _{n}^{i} of simply parallel geodesics with respect to ℒ̃_{n}. Clearly,

⋃_{i=1}^{k(n)} ℒ̃ _{n}^{i} ⊂ ⋃_{j(I)} ℒ̃ _{n+1}^{j(I)}.

Following the procedure in the construction of (ℒ_{n}, σ_{n}) in the previous proposition, for any small ε(I) > 0, we can have a finite earthquake measure σ_{n+1}^{I} supported on a finite lamination ℒ_{n+1}^{I} which is a refinement of ℒ_{n} such that

(34)  ∥V_{}_{σ̃_{n+1}^{I}} − V_{σ_{n+1}^{I}}∥_{∞} < ε(I),

where σ̃_{n+1}^{I} is the measure σ̃_{n+1} restricted on ⋃_{j(I)} ℒ̃ _{n+1}^{j(I)} .

Let σ̃_{n+1}^{II} be the restriction of σ̃_{n+1} on ⋃_{j(ii)} ℒ̃ _{n+1}^{j(II)} . Repeating the same pro-

cedure in the construction of (ℒ_{n}, σ_{n}) for (⋃_{j(II)} ℒ̃ _{n+1}^{j(II)}, σ̃_{n+1}^{II}), for any ε(II) > 0, we can obtain a finite earthquake measure (ℒ_{n+1}^{II}, σ_{n+1}^{II}) such that

(35)  ∥V_{}_{σ̃_{n+1}^{II}} − V_{σ_{n+1}^{II}}∥_{∞} < ε(II).

260 F. P. Gardiner and J. Hu

Let ℒ_{n+1} = ℒ_{n+1}^{I} ∪ ℒ_{n+1}^{II} and σ_{n+1} = σ_{n+1}^{I} + σ_{n+1}^{II}. Clearly, ℒ_{n+1} ⊃ ℒ_{n}. With the same reason to see ∥σ_{n}∥_{Th} ≤ 2∥σ̃_{n}∥_{Th} , we also have ∥σ_{n+1}∥_{Th} ≤ 2∥σ̃_{n+1}∥_{Th} . Given any ε_{n+1} > 0, let ε(I) = ε(II) = \frac{1}{2}ε_{n+1} . By (34) and (35), we obtain

∥Vσ̃_{n+1} − V_{σ_{n+1}}∥_{∞} < \frac{1}{n}.

It completes the proof.

Let ε_{n} = 1/n. Then Propositions 6, 7 and 8 imply our Theorem 2.

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Received 27 April 2004