Lipschitz images with fractal boundaries and their small surface wrapping

by Zoltán Buczolich

ABSTRACT

Assume E ⊂ H ⊂ R^{m} and Φ : E → R^{m} is a Lipschitz-L mapping; |H|, and ||H|| denotes the volume and the surface area of H. We verify that there exists a figure F ⊃ Φ(E) with ||F || ≤ c_{L}||H||, and, of course, |F | ≤ c_{L}|H|, where c_{L} depends only on the dimension and on L. We also give an example when E = H ⊂ R^{2} is a square and ||Φ(E)|| = ∞, in fact, the boundary of Φ(E) can contain a fractal of Hausdorff dimension exceeding one.

Introduction

Related to a question of W. F. Pfeffer about bi-Lipschitz transformations of certain generalized integrals we had to deal with the following problem. Assume that the set E ⊂ R^{m} is a subset of a set H ⊂ R^{m} with m-dimensional measure |H|, and with surface area ||H||, that is, the m − 1-dimensional Hausdorff measure of the essential boundary of H equals ||H||. One can say that E is wrapped into (the boundary of) H. The question which motivated this paper is the following: If Φ : E → R^{m} is a Lipschitz L-transformation then is it true that one can wrap Φ(E) into a set which has “not much more” surface area, and volume than that of H? To be more precise, is it true that there exists a constant c_{L} depending only on the dimension and on the Lipschitz constant L such that we can find a set F containing Φ(E) and satisfying ||F || ≤ c_{L}||H||, and |F | ≤ c_{L}|H|. In Theorem 1 we give an affirmative answer to this question. The first natural idea to prove Theorem 1 would be the usage of Kirszbraun’s theorem [Fe,Th. 2.10.43.] to extend the Lipschitz

Research supported by the Hungarian National Foundation for Scientific Research, Grant Nos. T 019476 and T 016094.

1991 Mathematics Subject Classification: Primary 28A75, Secondary 28A80, 26B35. Keywords: Lipschitz mapping, surface, fractal.

L-mapping Φ onto R^{m} and use F = Φ(H). However this simple idea does not seem to work. In Theorem 2 we show that even in the simple case when Q_{0} ⊂ R^{2} is a square in the plane there exists a Lipschitz mapping Φ_{0} : R^{2} → R^{2}, such that the perimeter of Φ_{0}(Q_{0}) is of infinite measure. In fact, the boundary of Φ_{0}(Q_{0}) can contain a fractal, a self-similar set of Hausdorff dimension bigger than one.

The proof of Theorem 1 is not difficult. Considering a suitable covering of H by open balls and the Φ image of this covering one can find the requiered figure. In our argument Besicovitch’s covering theorem and the relative isoperimetric inequality are used. We thank the referee for pointing out that the less known covering theorem [Fe, Corollary 4.5.4] could provide an even simpler argument, however to keep our paper self contained we use the few lines longer alternate argument, which also contains the nice, and not widely known, key idea of [Fe, Corollary 4.5.4].

Interpreting BV sets (sets of bounded variation) as integral currents [Fe, Section 4.5.1] and applying the area theorem [Fe, Theorem 3.2.3] and [Fe, Corollary 4.1.26] it is easy to see that a one-to-one Lipschitz image of a BV set is again BV, that is, one-to-one Lipschitz images of sets of finite surface area are again of finite surface area. The mapping Φ, constructed in the proof of Theorem 2 can be chosen to be conjugate (via a homeomorphism) to a folding of the square Q_{0}. Thus Φ is a two-to-one mapping at almost all points of Q_{0}, and it is one-to-one on the “folding” edge, which contains a fractal.

Preliminaries

If x, y are points of the Euclidean space R^{m} then the (Euclidean) distance of the points is denoted by |x − y|. The transformation Φ : R^{m} → R^{m} is a Lipschitz L-mapping if

|Φ(x) − Φ(y)| ≤ L|x − y|

holds for all x, y ∈ R^{m}. The open ball centered at x and of radius r will be denoted by B(x, r).

If H is a subset of R^{m} then |H| will denote its m-dimensional Lebesgue measure. (Note that for ease of notation the norm and the measure is denoted by the same symbol, though the domains of these symbols are different.) The m − 1-dimensional Hausdorff measure of H will be denoted by H(H).

A figure is a finite union of compact nondegenerate subintervals of R^{m}.

Assume that H ⊂ R^{m} is measurable. The set of its Lebesgue density points is called the essential interior of H and is denoted by int^{∗}H, that is,

int^{∗}H = \{x ∈ R^{m} : lim_{ϵ→0+} \frac{|H ∩ B(x, ϵ)|}{|B(x, ϵ)|} = 1\} .

The set cl^{∗}H = R^{m} \ int^{∗}(R^{m} \ H) is called the essential closure of H. Finally, ∂^{∗}H = cl^{∗}H \ int^{∗}H denotes the essential boundary of H. A set H ⊂ R^{m} is a BV set, a set of bounded variation, if it is bounded, measurable, and its perimeter ||H|| = H(∂^{∗}H) is finite. The family of BV sets arises quite often in geometric integration theory as the largest class where a reasonable surface integral can be defined, see, for example, [P, Section 2] where further references can be found as well. Clearly, figures are very simple sort of BV sets.

Recall from [M, Th. 2.7., p.30]

Besicovitch’s covering theorem. There is a contant c_{B} depending only on the dimension m with the following property. Let E be a bounded subset of R^{m}, and let B be a family of balls such that each point of E is the centre of some ball of B. Then there are families B_{1}, ..., B_{c_{B}} covering E such that each B_{i} is disjoint, that is,

E ⊂ ⋃_{i} ⋃ B_{i},

and

B ∩ B′ = ∅ for B, B′ ∈ B_{i} with B ≠ B′.

Besicovitch’s covering theorem is stated in [M] for families of closed balls. Scrutinizing its proof one can easily see that the theorem holds for familes of open balls as well.

We will also use from [Z, Theorem 5.4.3., p.230] the

Relative isoperimetric inequality. If H ⊂ R^{m} a BV -set then for each ball B(x, r) ⊂ R^{m} we have

min\{|B(x, r) ∩ H|, |B(x, r) \ H|\}^{\frac{m−1}{m}} ≤ c_{iso}H(B(x, r) ∩ ∂^{∗}H),

where the constant c_{iso} depends only on the dimension.

Next we remind to some properties of self-similar sets. For the details see, for example, [Fa, Ch.9., p.113-137]. Assume that the mappings φ_{j} : R^{m} → R^{m}, (j = 1, ..., k) are similarities with similarity ratios c_{j}. The (measurable) set F is self similar if

F = ⋃_{j=1}^{k} φ_{j}(F).

The mappings φ_{j} satisfy the open set condition if there exists a non-empty bounded open set such that

V ⊃ ⋃_{j=1}^{k} φ_{j}(V),

with the union disjoint. If the mappings φ_{j} satisfy the open set condition then the Hausdorff dimension of the self similar set F, denoted by s, is given by

∑_{j=1}^{k} c_{j}^{s} = 1.

In the special case when all c_{j}’ s have the same value, say c, then s = log(1/k)/ log c.

Main Results

Theorem 1. Assume that E ⊂ R^{m} is compact, Φ : E → R^{m} is a Lipschitz Lmapping, and H is a BV set such that E ⊂ int^{∗}H. Then there exists a figure F, and a constant c_{L}, depending only on L and on the dimension, such that Φ(E) ⊂ F, ||F || ≤ c_{L}||H||, and |F | ≤ c_{L}|H|.

Proof. For x ∈ E put

g_{x}(r) = \frac{|B(x, r) ∩ H|}{|B(x, r)|}.

Since x ∈ int^{∗}H we have lim_{r→0+} g_{x}(r) = 1. On the other hand H is bounded; hence lim_{r→∞} g_{x}(r) = 0. Since g_{x}(r) is continuous we can choose r_{x}, such that g_{x}(r_{x}) = 1/2.

Using Besicovitch’s covering theorem choose c_{B} many classes {B_{i}}, each consisting of balls B(x, r_{x}), such that

E ⊂ ⋃_{i} ⋃_{B(x,r_{x})∈B_{i}} B(x, r_{x}),

and balls belonging to the same class B_{i} are disjoint. Since E is compact we can also assume that each B_{i} consists of finitely many balls.

Using the relative isoperimetric inequality and g_{x}(r_{x}) = 1/2, we have

(|B(x, r_{x})|/2)^{\frac{m−1}{m}} ≤ c_{iso}H(B(x, r_{x}) ∩ ∂^{∗}H)

for each B(x, r_{x}), that is, there exists a constant c′_{iso} such that

||B(x, r_{x})|| ≤ c′_{iso}H(B(x, r_{x}) ∩ ∂^{∗}H).

Using that balls belonging to a class B_{i} are disjoint, we have

∑_{B(x,r_{x})∈B_{i}} ||B(x, r_{x})|| ≤ c′_{iso} ∑_{B(x,r_{x})∈B_{i}} H(B(x, r_{x}) ∩ ∂^{∗}H) ≤ c′_{iso}||H||,

and

∑_{B(x,r_{x})∈B_{i}} |B(x, r_{x})| = 2 ∑_{B(x,r_{x})∈B_{i}} |B(x, r_{x}) ∩ H| ≤ 2|H|,

for each i ≤ c_{B}. Clearly, Φ(B(x, r_{x})) ⊂ B(Φ(x), Lr_{x}) ⊂ Q_{Φ(x)}, where Q_{Φ(x)} is a closed cube centered at Φ(x) and of side length 2Lr_{x}. Then there exists a constant c_{Q}, depending only on the dimension and on L, such that ||Q_{Φ(x)}|| ≤ c_{Q}||B(x, r_{x})||, and |Q_{Φ(x)}| ≤ c_{Q}|B(x, r_{x})|.

Let

F = ⋃_{i} ⋃_{B(x,r_{x})⊂B_{i}} Q_{Φ(x)}.

Then

||F || ≤ ∑_{i} ∑_{B(x,r_{x})⊂B_{i}} ||Q_{Φ(x)}|| ≤ c_{B}c_{Q}c′_{iso}||H||,

|F | ≤ ∑_{i} ∑_{B(x,r_{x})⊂B_{i}} |Q_{Φ(x)}| ≤ 2c_{B}c_{Q}|H|,

and clearly Φ(E) ⊂ F. Therefore, one can define c_{L} as the maximum of the constants c_{B}c_{Q}c′_{iso} and 2c_{B}c_{Q}. This completes the proof.

Theorem 2. There exists a Lipschitz transformation Φ_{0} : R^{2} → R^{2} and a square Q_{0} ⊂ R^{2} such that the Hausdorff dimension of ∂^{∗}(Φ_{0}(Q_{0})) is bigger than one and hence, it is of non-σ-finite linear measure. Furthermore Φ_{0} is two-to-one at almost all points of Q_{0} and is one-to-one on the rest of Q_{0}.

Proof. It is enough to define a suitable Lipschitz transformation Φ on a rectangle Q_{1} with vertices (0, 3), (48, 3), (48, −3), and (0, −3), since then, by using a suitable affine transformation, we can define Φ_{0} on Q_{0}, and then by using Kirszbraun’s theorem we can extend Φ_{0} onto R^{2}.

We construct Φ by a limit of a self-similar like repetition of the same mapping.

First we define a Lipschitz folding Φ_{1} of Q_{1}. Then we define 48 subrectangles Q_{2}^{1},...,Q_{2}^{48} in Q_{1}, such that each Q_{2}^{j} is similar to Q_{1}. Our mapping Φ will coincide with Φ_{1} on Q_{1} \ ∪_{j=1}^{48}Q_{2}^{j}. On the rectangles Q_{2}^{j} we will do a folding process similar to the one in Q_{1}.

Put s_{1} = (0, 3), s_{2} = (1, 3), s_{3} = (2, 3), s_{4} = (2, −3), s_{5} = (1, −3), and s_{6} = (0, −3).

First we define Φ_{1} on the rectangle S_{1} = s_{1}s_{2}s_{5}s_{6}.

Choose the points p_{1}, p_{2}, p_{3}, and p_{4} on the line y = x such that p_{1} = (0, 0), p_{2} = (0.1, 0.1), p_{3} = (0.9, 0.9), and p_{4} = (1, 1). Next we choose the points q_{1}, q_{2}, q_{3}, q_{4}, and r_{1}, r_{2}, r_{3}, r_{4} on the lines y = x−0.1, and y = x+0.1, respectively; such that q_{1} = (0, −0.1), r_{1} = (0, 0.1), q_{2} = (0.15, 0.05), r_{2} = (0.05, 0.15), q_{3} = (0.95, 0.85), r_{3} = (0.85, 0.95), q_{4} = (1, 0.9), and r_{4} = (1, 1.1).

Denote by Q_{2}^{1} the rectangle q_{2}q_{3}r_{3}r_{2}. Observe that Q_{2}^{1} is similar to Q_{1}.

Let Φ_{1} equal the identity on the segments p_{1}s_{6} and s_{6}s_{5}, and equal the reflection about the x-axis on the segments p_{1}s_{1} and s_{1}s_{2}. On r_{2}p_{2}p_{3}r_{3}, that is, on the upper half of Q_{2}^{1} we define Φ_{1} to be equal to the reflection about the y = x line; on the lower half of Q_{2}^{1}, that is, on p_{2}q_{2}q_{3}p_{3} we define Φ_{1} to be equal to the identity, that is, Φ_{1} is a lengthwise folding on Q_{2}^{1}.

At points of S_{1}, where we have not already defined Φ_{1}, we extend its definition to meet the following requirements:

i) Φ_{1} is Lipschitz on S_{1}; ii) Φ_{1} maps S_{1} onto the trapezoid p_{1}s_{6}s_{5}p_{4}, which will be denoted by T_{1}; iii) Φ_{1} is one-to-one on the segment p_{1}p_{4} and is two-to-one at any other point of S_{1}; hence

Φ_{1}(S_{1} \ Q_{2}^{1}) = T_{1} \ Q_{2}^{1}; iv) Φ_{1} maps the line segment s_{2}p_{4} onto the segment p_{4}s_{5};

v) Φ_{1} equals the identity on the segments p_{4}s_{5}, p_{1}p_{4}. It is an easy exercise, left to the reader, to show that the above conditions can be

satisfied.

Next we define Φ_{1} on the strip S_{2} = s_{2}s_{3}s_{4}s_{5}. Denote by τ the reflection about the line x = 1. For p ∈ S_{2} set Φ_{1}(p) = τ(Φ_{1}(τ(p))). Put Q_{2}^{2} = τ(Q_{2}^{1}). Then Q_{2}^{2} is a rectangle similar to Q_{1}, and Φ_{1} on Q_{2}^{2} equals a lengthwise folding of Q_{2}^{2}. Furthermore letting T_{2} = τ(T_{1}) we have the following properties satisfied.

i) Φ_{1} is Lipschitz on S_{1} ∪ S_{2}; ii) Φ_{1} maps S_{1} ∪ S_{2} onto T_{1} ∪ T_{2}; iii) Φ_{1} is one-to-one on the segments p_{1}p_{4} and p_{4}τ(p_{1}), and is two-to-one at any other

point of S_{1} ∪ S_{2}; hence Φ_{1}((S_{1} ∪ S_{2}) \ (Q_{2}^{1} ∪ Q_{2}^{2})) = (T_{1} ∪ T_{2}) \ (Q_{2}^{1} ∪ Q_{2}^{2}); iv) Φ_{1} equals the identity on the segments p_{1}s_{6}, s_{6}s_{4}, s_{4}τ(p_{1}) and Φ_{1} equals the reflection

about the x-axis on the segments p_{1}s_{1}, s_{1}s_{3}, and s_{3}τ(p_{1}), that is, on the boundary of S_{1} ∪ S_{2} the mapping Φ_{1} coincides with a folding about the x-axis. Denote by σ_{j} the translation by the vector (0, 2j). The strip bounded by the points

(j−1, 3), (j, 3), (j, −3), and (j−1, −3) is denoted by S_{j}. If p ∈ S_{2j−1}∪S_{2j}, (j = 2, 3, ..., 24) then put Φ_{1}(p) = σ_{j−1}(Φ_{1}(σ_{j−1}^{−1}(p))), Q_{2}^{2j−1} = σ_{j−1}(Q_{2}^{1}), and Q_{2}^{2j} = σ_{j−1}(Q_{2}^{2}). Then one can easily see that Φ_{1} satisfies the following properties:

i) Φ_{1} is Lipschitz on Q_{1}, denote its Lipschitz constant by L; ii) Φ_{1} coincides with a lengthwise folding on the boundary of Q_{1}, and on the whole

rectangles Q_{2}^{j}, (j = 1, ..., 48); iii) If g(x) denotes the sawtooth function which is periodic by 2, equals y = x on [0, 1],

and y = −x + 2 on [1, 2], then Φ_{1}(Q_{1}) is the region bounded by the graph of g and by the lower half of the boundary of Q_{1};

iv) Φ_{1} is one-to-one on the part of the boundary of Φ_{1}(Q_{1}) which is on the graph of g; at any other point of Q_{1} it is two-to-one. Denote by φ_{j} the similarities which map Q_{1} onto Q_{2}^{j}, (j = 1, ..., 48), and the lower

half of Q_{1} is mapped onto the lower half of Q_{2}^{j}. Obseve that the contraction ratio of φ_{j} equals \sqrt{2}/60.

For p ∈ Q_{2}^{j} put Φ_{2}^{j}(p) = φ_{j}(Φ_{1}(φ_{j}^{−1}(p))). Then Φ_{2}^{j} is a Lipschitz L-mapping, and it coincides with Φ_{1} on the boundary of Q_{2}^{j}, since on this boundary they both equal a lengthwise folding of Q_{2}^{j}.

Put Φ̃_{1}(p) = Φ_{1}(p) if p ∈ Q_{1} \ ∪_{j=1}^{48}Q_{2}^{j}, and Φ̃_{1}(p) = Φ_{2}^{j}(p) if p ∈ Q_{2}^{j}. Then Φ̃_{1} is a Lipschitz L-mapping. Indeed, assume p, q ∈ Q_{1}. If p, q ∉ ∪_{j=1}^{48}Q_{2}^{j}, or

there exists j such that p, q ∈ Q_{2}^{j} then |Φ̃_{1}(p) − Φ̃_{1}(q)| ≤ L|p − q| follows from the same property for Φ_{1}, or for Φ_{2}^{j}.

If p ∈ Q_{1}, and q ∈ Q_{2}^{j} then the segment pq intersects the boundary of Q_{2}^{j}. Denote this intersection point by q^{∗}. Then

|Φ̃_{1}(p) − Φ̃_{1}(q)| ≤ |Φ̃_{1}(p) − Φ̃_{1}(q^{∗})| + |Φ̃_{1}(q^{∗}) − Φ̃_{1}(q)| =
|Φ_{1}(p) − Φ_{1}(q^{∗})| + |Φ_{2}^{j}(q^{∗}) − Φ_{2}^{j}(q)| ≤ L(|p − q^{∗}| + |q^{∗} − q|) = L|p − q|.

Finally if p ∈ Q_{2}^{i} and q ∈ Q_{2}^{j} then denoting by p^{∗} and q^{∗} the intersection points of pq and the boundaries of Q_{2}^{i} and Q_{2}^{j}, respectively, an argument, similar to the one in the above paragraph, can establish the Lipschitz L-property of Φ̃_{1}.

It is also easy to see that Φ̃_{1} is two-to-one at almost all points of Q_{1}, the exceptional points are on the “folding edges” and at these points Φ̃_{1} is one-to-one.

Assume k ≥ 2 and j_{1}, ..., j_{k} ∈ {1, ..., 48} and p ∈ Q_{1}. Put φ_{j_{1},...,j_{k}}(p) = φ_{j_{1}} ∘...∘φ_{j_{k}}(p), Q_{k+1}^{j_{1},...,j_{k}} = φ_{j_{1},...,j_{k}}(Q_{1}), and for p ∈ Q_{k+1}^{j_{1},...,j_{k}}, let Φ_{k+1}^{j_{1},...,j_{k}}(p) = φ_{j_{1},...,j_{k}}(Φ_{1}(φ_{j_{1},...,j_{k}}^{−1}(p))). Denote Q′_{k+1} = ∪_{j_{1},...,j_{k}∈{1,...,48}}Q_{k+1}^{j_{1},...,j_{k}}.

If Φ̃_{k−1}, (k = 2, ...) is given then put Φ̃_{k}(p) = Φ̃_{k−1}(p), for p ∈ Q_{1} \ Q′_{k+1}, and Φ̃_{k}(p) = Φ_{k+1}^{j_{1},...,j_{k}}(p), for p ∈ Q_{k+1}^{j_{1},...,j_{k}}. Then, it is not difficult to see, that Φ̃_{k} is a Lipschitz L-mapping and is a lengthwise folding on the boundaries of Q_{k+1}^{j_{1},...,j_{k}} and on the sets Q_{k+2}^{j_{1},...,j_{k+1}} = φ_{j_{1},...,j_{k},j_{k+1}}(Q_{1}).

Since diam(Q_{k+1}^{j_{1},...,j_{k}}) → 0 as k → ∞ it is easy to see that the sequence Φ̃_{k} uniformly converges on Q_{1} to a Lipschitz L-mapping which we will denote by Φ.

Put Q_{∞} = ∩_{k=1}^{∞}Q′_{k+1}. Since Φ(Q_{1}) contains a little more than half of each Q_{k+1}^{j_{1},...,j_{k}}, examining our self-similar construction and using the definition of Φ on Q_{k+1}^{j_{1},...,j_{k}}, one can easily see that points of Q_{∞} belong to ∂^{∗}(Φ(Q_{1})).

Since Q_{∞} = ∪_{j=1}^{48}φ_{j}(Q_{∞}), taking an open set V, which is slightly larger than Q_{1}, the open set condition also holds. Using that the φ_{j}’s have contraction ratio \sqrt{2}/60 we infer that

dim_{H}(∂^{∗}(Φ(Q_{1}))) ≥ dim_{H}(Q_{∞}) = \frac{log(\frac{1}{48})}{log(\frac{\sqrt{2}}{60})} > 1.03.

Finally, it is not difficult to see that Φ is two-to-one on Q_{1} everywhere but the points of the “folding edge”, where it is one-to-one. This completes our proof.

Remark. By modifying some constants in our construction and by using steeper sawtooth functions (for the internal folding edge in the definition of Φ_{1}) one can get the Hausdorff dimension of ∂^{∗}(Φ(Q_{1}))) arbitrarily close to 2.

REFERENCES

[M] P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge University Press, 1995.

[Fa] K. Falconer, Fractal Geometry, John Wiley & Sons, 1990.

[Fe] H. Federer, Geometric Measure Theory, Springer Verlag, New York, 1969.

[P] W.F. Pfeffer, The Gauss-Green theorem, Adv. Math., 87, 93-147, 1991.

[Z] W. P. Ziemer, Weakly differentiable functions, Springer Verlag, New York, 1989.

Eötvös Loránd University, Department of Analysis, Budapest, Múzeum krt 6-8, H-1088, Hungary.

email: buczo@cs.elte.hu