这是用户在 2025-6-6 15:45 为 https://app.immersivetranslate.com/pdf-pro/970078c0-ad25-4dea-a43c-881aea699a27/ 保存的双语快照页面，由 沉浸式翻译 提供双语支持。了解如何保存？

HAUSDORFF DIMENSION AND QUASICONFORMAL MAPPINGS
HAUSDORFF 维数与拟共形映射

F. W. GEHRING † † †\dagger AND J. VÄISÄLÄDedicated to the memory of A. S. Besicovitch
为纪念 A. S. Besicovitch

  1. Introduction. In this paper we study what happens to the Hausdorff dimension of a set A A AA, denoted by H - dim A dim A dim A\operatorname{dim} A, under an n n nn-dimensional quasiconformal mapping f : D → D ′ f : D → D ′ f:D rarrD^(')f: D \rightarrow D^{\prime} with A ⊂ D A ⊂ D A sub DA \subset D. It is clear that
    引言。在本文中，我们研究在 n n nn 维拟共形映射 f : D → D ′ f : D → D ′ f:D rarrD^(')f: D \rightarrow D^{\prime} 作用下，集合 A A AA 的 Hausdorff 维数 H - dim A dim A dim A\operatorname{dim} A 会发生什么。很明显，
H − dim f [ A ] = H − dim A H − dim f [ A ] = H − dim A H-dim f[A]=H-dim A\mathrm{H}-\operatorname{dim} f[A]=\mathrm{H}-\operatorname{dim} A
if f f ff is a diffeomorphism or, more generally, if f f ff and f − 1 f − 1 f^(-1)f^{-1} are locally Lipschitzian. We show first, however, that (1) need not hold if f f ff is a general quasiconformal mapping. Next we give bounds for H -dim f [ A ] f [ A ] f[A]f[A] in terms of H -dim A , n A , n A,nA, n, and the maximal dilatation of f f ff. In particular, we prove that H - dim A = 0 implies H dim A = 0 implies H dim A=0impliesH\operatorname{dim} A=0 \mathrm{implies} \mathrm{H} - dim f [ A ] = 0 dim f [ A ] = 0 dim f[A]=0\operatorname{dim} f[A]=0, and we conjecture that H − dim A = n H − dim A = n H-dim A=n\mathrm{H}-\operatorname{dim} A=n implies H − dim f [ A ] = n H − dim f [ A ] = n H-dim f[A]=n\mathrm{H}-\operatorname{dim} f[A]=n, or equivalently that H - dim A < n dim A < n dim A < n\operatorname{dim} A<n implies H - dim f [ A ] < n dim f [ A ] < n dim f[A] < n\operatorname{dim} f[A]<n. We establish this conjecture for the case where n = 2 n = 2 n=2n=2 and then prove that, for general n , H n , H n,Hn, \mathrm{H} - dim f [ A ] < n dim f [ A ] < n dim f[A] < n\operatorname{dim} f[A]<n whenever A A AA is contained in an m m mm-dimensional hyperplane with m < n m < n m < nm<n. An example shows that H-dim f [ A ] f [ A ] f[A]f[A] can be arbitrarily close to n n nn, even when A A AA is a 1 -dimensional segment.
如果 f f ff 是一个微分同胚，或者更一般地，如果 f f ff 和 f − 1 f − 1 f^(-1)f^{-1} 是局部 Lipschitz 的。然而，我们首先证明，如果 f f ff 是一个一般的拟共形映射，则 (1) 不一定成立。接下来，我们给出 H-维数 f [ A ] f [ A ] f[A]f[A] 关于 H-维数 A , n A , n A,nA, n 和 f f ff 的最大扩张的界限。特别地，我们证明了 H- dim A = 0 implies H dim A = 0 implies H dim A=0impliesH\operatorname{dim} A=0 \mathrm{implies} \mathrm{H} - dim f [ A ] = 0 dim f [ A ] = 0 dim f[A]=0\operatorname{dim} f[A]=0 ，并猜想 H − dim A = n H − dim A = n H-dim A=n\mathrm{H}-\operatorname{dim} A=n 蕴含 H − dim f [ A ] = n H − dim f [ A ] = n H-dim f[A]=n\mathrm{H}-\operatorname{dim} f[A]=n ，或者等价地，H- dim A < n dim A < n dim A < n\operatorname{dim} A<n 蕴含 H- dim f [ A ] < n dim f [ A ] < n dim f[A] < n\operatorname{dim} f[A]<n 。我们为 n = 2 n = 2 n=2n=2 的情况建立了这个猜想，然后证明，对于一般的 n , H n , H n,Hn, \mathrm{H} - dim f [ A ] < n dim f [ A ] < n dim f[A] < n\operatorname{dim} f[A]<n ，只要 A A AA 包含在一个 m m mm 维的超平面中，并且 m < n m < n m < nm<n 。一个例子表明，即使当 A A AA 是一个一维线段时，H-dim f [ A ] f [ A ] f[A]f[A] 也可以任意接近 n n nn 。
2. Notation. We shall use the terminology and notation for quasiconformal mappings given in [16]. Moreover, since we are concerned only with local properties which are invariant under Möbius transformations, we shall consider only quasiconformal mappings f : D → D ′ f : D → D ′ f:D rarrD^(')f: D \rightarrow D^{\prime} where D D DD and D ′ D ′ D^(')D^{\prime} are domains in the non-compact n n nn-dimensional Euclidean space R n R n R^(n)R^{n}.
2. 符号说明。我们将使用文献[16]中给出的拟共形映射的术语和符号。此外，由于我们只关心在 Möbius 变换下保持不变的局部性质，我们将只考虑拟共形映射 f : D → D ′ f : D → D ′ f:D rarrD^(')f: D \rightarrow D^{\prime} ，其中 D D DD 和 D ′ D ′ D^(')D^{\prime} 是非紧致 n n nn 维欧几里得空间 R n R n R^(n)R^{n} 中的区域。
For a ∈ ( 0 , ∞ ) a ∈ ( 0 , ∞ ) a in(0,oo)a \in(0, \infty), the Hausdorff a-dimensional outer measure of a set A ⊂ R n A ⊂ R n A subR^(n)A \subset R^{n} is defined as
对于 a ∈ ( 0 , ∞ ) a ∈ ( 0 , ∞ ) a in(0,oo)a \in(0, \infty) ，集合 A ⊂ R n A ⊂ R n A subR^(n)A \subset R^{n} 的 Hausdorff a 维外测度定义为
H a ( A ) = lim d → 0 ( inf ∑ i dia ( A i ) a ) , H a ( A ) = lim d → 0   inf ∑ i   dia A i a , H_(a)(A)=lim_(d rarr0)(i n fsum_(i)dia(A_(i))^(a)),H_{a}(A)=\lim _{d \rightarrow 0}\left(\inf \sum_{i} \operatorname{dia}\left(A_{i}\right)^{a}\right),
where the infimum is taken over all countable coverings of A A AA by sets A i A i A_(i)A_{i} with dia ( A i ) < d dia A i < d dia(A_(i)) < d\operatorname{dia}\left(A_{i}\right)<d. The Hausdorff dimension of A A AA is then given by
其中，infimum 是在所有用满足 dia ( A i ) < d dia A i < d dia(A_(i)) < d\operatorname{dia}\left(A_{i}\right)<d 的集合 A i A i A_(i)A_{i} 对 A A AA 进行可数覆盖的情况下取的。然后， A A AA 的 Hausdorff 维数为
H − dim A = inf { a : H a ( A ) = 0 } . H − dim A = inf a : H a ( A ) = 0 . H-dim A=i n f{a:H_(a)(A)=0}.\mathrm{H}-\operatorname{dim} A=\inf \left\{a: H_{a}(A)=0\right\} .
Clearly 0 ⩽ H 0 ⩽ H 0 <= H0 \leqslant \mathrm{H}-dim A ⩽ n A ⩽ n A <= nA \leqslant n.
显然 0 ⩽ H 0 ⩽ H 0 <= H0 \leqslant \mathrm{H} -dim A ⩽ n A ⩽ n A <= nA \leqslant n 。
3. We shall need the following generalization of a result due to Mori [13; Lemma 4].
3. 我们需要 Mori [13; 引理 4] 的一个推广结果。
Lemma. Suppose that f : D → D ′ f : D → D ′ f:D rarrD^(')f: D \rightarrow D^{\prime} is an n n nn-dimensional K K KK-quasiconformal mapping, that U U UU is a bounded domain with U ¯ ⊂ D U ¯ ⊂ D bar(U)sub D\bar{U} \subset D, and that x ∈ U x ∈ U x in Ux \in U. Let
引理。假设 f : D → D ′ f : D → D ′ f:D rarrD^(')f: D \rightarrow D^{\prime} 是一个 n n nn 维的 K K KK 拟共形映射， U U UU 是一个有界域，且 U ¯ ⊂ D U ¯ ⊂ D bar(U)sub D\bar{U} \subset D ，并且 x ∈ U x ∈ U x in Ux \in U 。令
M = max y ∈ ∂ U | y − x | , m = min y ∈ ∂ U | y − x | , L = max y ∈ ∂ U | f ( y ) − f ( x ) | , l = min y ∈ ∂ U | f ( y ) − f ( x ) | M = max y ∈ ∂ U   | y − x | , m = min y ∈ ∂ U   | y − x | , L = max y ∈ ∂ U   | f ( y ) − f ( x ) | , l = min y ∈ ∂ U   | f ( y ) − f ( x ) | M=max_(y in del U)|y-x|,m=min_(y in del U)|y-x|,L=max_(y in del U)|f(y)-f(x)|,l=min_(y in del U)|f(y)-f(x)|M=\max _{y \in \partial U}|y-x|, m=\min _{y \in \partial U}|y-x|, L=\max _{y \in \partial U}|f(y)-f(x)|, l=\min _{y \in \partial U}|f(y)-f(x)|.
If the ball B n ( f ( x ) , L ) B n ( f ( x ) , L ) B^(n)(f(x),L)B^{n}(f(x), L) is contained in D ′ D ′ D^(')D^{\prime}, then
如果球 B n ( f ( x ) , L ) B n ( f ( x ) , L ) B^(n)(f(x),L)B^{n}(f(x), L) 包含在 D ′ D ′ D^(')D^{\prime} 中，那么
L ⩽ C l , L ⩽ C l , L <= Cl,L \leqslant C l,
where C C CC is a finite constant which depends only on n , K n , K n,Kn, K, and M / m M / m M//mM / m.
其中 C C CC 是一个仅依赖于 n , K n , K n,Kn, K 和 M / m M / m M//mM / m 的有限常数。
Proof. Suppose that l < L l < L l < Ll<L and let R R RR denote the image under f − 1 f − 1 f^(-1)f^{-1} of the spherical ring
证明。假设 l < L l < L l < Ll<L ，并让 R R RR 表示球环在 f − 1 f − 1 f^(-1)f^{-1} 下的像
R ′ = { y : l < | y − f ( x ) | < L } ⊂ D ′ . R ′ = { y : l < | y − f ( x ) | < L } ⊂ D ′ . R^(')={y:l < |y-f(x)| < L}subD^(').R^{\prime}=\{y: l<|y-f(x)|<L\} \subset D^{\prime} .
Then R R RR is a ring which separates x x xx and a point y ∈ ∂ U y ∈ ∂ U y in del Uy \in \partial U from ∞ ∞ oo\infty and a point z ∈ ∂ U z ∈ ∂ U z in del Uz \in \partial U. Hence if Γ Γ Gamma\Gamma is the family of arcs joining the components of C ( R ) C ( R ) C(R)C(R) in R R RR, it follows from the extremal property of the Teichmüller ring in R n [ 4 R n [ 4 R^(n)[4R^{n}[4 and 14], or from [16; 11.9] that
那么 R R RR 是一个环，它将 x x xx 和点 y ∈ ∂ U y ∈ ∂ U y in del Uy \in \partial U 与 ∞ ∞ oo\infty 和点 z ∈ ∂ U z ∈ ∂ U z in del Uz \in \partial U 分开。因此，如果 Γ Γ Gamma\Gamma 是连接 C ( R ) C ( R ) C(R)C(R) 中 R R RR 的成分的弧族，那么根据 Teichmüller 环在 R n [ 4 R n [ 4 R^(n)[4R^{n}[4 和 14]中的极值性质，或者根据[16; 11.9]，可以得出
M ( Γ ) ⩾ h n ( | z − x | | y − x | ) ⩾ h n ( M m ) M ( Γ ) ⩾ h n | z − x | | y − x | ⩾ h n M m M(Gamma) >= h_(n)((|z-x|)/(|y-x|)) >= h_(n)((M)/(m))M(\Gamma) \geqslant h_{n}\left(\frac{|z-x|}{|y-x|}\right) \geqslant h_{n}\left(\frac{M}{m}\right)
where h n : ( 0 , ∞ ) → ( 0 , ∞ ) h n : ( 0 , ∞ ) → ( 0 , ∞ ) h_(n):(0,oo)rarr(0,oo)h_{n}:(0, \infty) \rightarrow(0, \infty) is positive and decreasing. Then since f f ff is K K KK-quasiconformal
其中 h n : ( 0 , ∞ ) → ( 0 , ∞ ) h n : ( 0 , ∞ ) → ( 0 , ∞ ) h_(n):(0,oo)rarr(0,oo)h_{n}:(0, \infty) \rightarrow(0, \infty) 是正的且递减的。然后由于 f f ff 是 K K KK -拟共形的
M ( Γ ) ⩽ K M ( f [ Γ ] ) = K ω n − 1 ( log ⁡ L l ) 1 − n M ( Γ ) ⩽ K M ( f [ Γ ] ) = K ω n − 1 log ⁡ L l 1 − n M(Gamma) <= KM(f[Gamma])=Komega_(n-1)(log((L)/(l)))^(1-n)M(\Gamma) \leqslant K M(f[\Gamma])=K \omega_{n-1}\left(\log \frac{L}{l}\right)^{1-n}
and (4) follows from (5) and (6).
并且(4)由(5)和(6)得出。
4. The Cantor sets C s n C s n C_(s)^(n)C_{s}{ }^{n}. For each integer n ⩾ 1 n ⩾ 1 n >= 1n \geqslant 1 and each s ∈ ( 0 , 1 2 ) s ∈ 0 , 1 2 s in(0,(1)/(2))s \in\left(0, \frac{1}{2}\right) we define a family of Cantor sets C s n C s n C_(s)^(n)C_{s}{ }^{n} as follows. Let Q Q QQ denote the closed unit cube
4. 康托集 C s n C s n C_(s)^(n)C_{s}{ }^{n} 。对于每个整数 n ⩾ 1 n ⩾ 1 n >= 1n \geqslant 1 和每个 s ∈ ( 0 , 1 2 ) s ∈ 0 , 1 2 s in(0,(1)/(2))s \in\left(0, \frac{1}{2}\right) ，我们定义一个康托集族 C s n C s n C_(s)^(n)C_{s}{ }^{n} 如下。让 Q Q QQ 表示闭单位立方体
Q = { x = ( x 1 , … , x n ) : 0 ⩽ x i ⩽ 1 } , Q = x = x 1 , … , x n : 0 ⩽ x i ⩽ 1 , Q={x=(x_(1),dots,x_(n)):0 <= x_(i) <= 1},Q=\left\{x=\left(x_{1}, \ldots, x_{n}\right): 0 \leqslant x_{i} \leqslant 1\right\},
choose a collection of 2 n 2 n 2^(n)2^{n} disjoint closed cubes Q i Q i Q_(i)Q_{i} of side s s ss in int Q , 1 ⩽ i ⩽ 2 n Q , 1 ⩽ i ⩽ 2 n Q,1 <= i <= 2^(n)Q, 1 \leqslant i \leqslant 2^{n}, oriented so that for each i i ii there exists a similarity mapping
选择一组 2 n 2 n 2^(n)2^{n} 个互不相交的闭立方体 Q i Q i Q_(i)Q_{i} ，边长为 s s ss ，位于整数集 Q , 1 ⩽ i ⩽ 2 n Q , 1 ⩽ i ⩽ 2 n Q,1 <= i <= 2^(n)Q, 1 \leqslant i \leqslant 2^{n} 中，方向使得对于每个 i i ii ，存在一个相似映射
g i ( x ) = s x + a i , a i ∈ Q g i ( x ) = s x + a i , a i ∈ Q g_(i)(x)=sx+a_(i),a_(i)in Qg_{i}(x)=s x+a_{i}, a_{i} \in Q
which maps Q Q QQ onto Q i Q i Q_(i)Q_{i}. Such collections of cubes Q i Q i Q_(i)Q_{i} obviously exist for each s ∈ ( 0 , 1 2 ) s ∈ 0 , 1 2 s in(0,(1)/(2))s \in\left(0, \frac{1}{2}\right). Next for each j ⩾ 1 j ⩾ 1 j >= 1j \geqslant 1 let
将 Q Q QQ 映射到 Q i Q i Q_(i)Q_{i} 。显然，对于每个 s ∈ ( 0 , 1 2 ) s ∈ 0 , 1 2 s in(0,(1)/(2))s \in\left(0, \frac{1}{2}\right) ，都存在这样的立方体集合 Q i Q i Q_(i)Q_{i} 。接下来，对于每个 j ⩾ 1 j ⩾ 1 j >= 1j \geqslant 1 ，令
F j = ⋃ i 1 , … , i j = 1 2 n g i 1 ∘ … ∘ g i j [ Q ] F j = ⋃ i 1 , … , i j = 1 2 n   g i 1 ∘ … ∘ g i j [ Q ] F_(j)=uuu_(i_(1),dots,i_(j)=1)^(2^(n))g_(i_(1))@dots@g_(i_(j))[Q]F_{j}=\bigcup_{i_{1}, \ldots, i_{j}=1}^{2^{n}} g_{i_{1}} \circ \ldots \circ g_{i_{j}}[Q]
Then { F j } F j {F_(j)}\left\{F_{j}\right\} is a decreasing sequence of compact sets, and each set F j F j F_(j)F_{j} is the union of 2 j n 2 j n 2^(jn)2^{j n} disjoint closed cubes of side s j s j s^(j)s^{j}. Hence
那么 { F j } F j {F_(j)}\left\{F_{j}\right\} 是一个紧集的递减序列，并且每个集合 F j F j F_(j)F_{j} 是 2 j n 2 j n 2^(jn)2^{j n} 个互不相交的边长为 s j s j s^(j)s^{j} 的闭立方体的并集。因此
C s n = ⋂ j = 1 ∞ F j C s n = ⋂ j = 1 ∞   F j C_(s)^(n)=nnn_(j=1)^(oo)F_(j)C_{s}^{n}=\bigcap_{j=1}^{\infty} F_{j}
is a compact set, and
是一个紧集，并且
H- dim C s n = n log ⁡ 1 2 log ⁡ s  H-  dim C s n = n log ⁡ 1 2 log ⁡ s " H- "dimC_(s)^(n)=n(log((1)/(2)))/(log s)\text { H- } \operatorname{dim} C_{s}^{n}=n \frac{\log \frac{1}{2}}{\log s}
by, for example, [1; Theorem 3] or [12; Theorem III]. In particular,
例如根据 [1; 定理 3] 或 [12; 定理 III] 可知。特别是
0 < H − dim C s . n < n 0 < H − dim C s . n < n 0 < H-dimC_(s_(.))^(n) < n0<\mathrm{H}-\operatorname{dim} C_{s_{.}}^{n}<n
and和
lim s → 0 H − dim C s n = 0 , lim s → 1 2 H − dim C s n = n lim s → 0   H − dim C s n = 0 , lim s → 1 2   H − dim C s n = n lim_(s rarr0)H-dimC_(s)^(n)=0,quadlim_(s rarr(1)/(2))H-dimC_(s)^(n)=n\lim _{s \rightarrow 0} \mathrm{H}-\operatorname{dim} C_{s}^{n}=0, \quad \lim _{s \rightarrow \frac{1}{2}} \mathrm{H}-\operatorname{dim} C_{s}^{n}=n
  1. Theorem. For each integer n ⩾ 2 n ⩾ 2 n >= 2n \geqslant 2 and each pair of such Cantor sets C s n C s n C_(s)^(n)C_{s}{ }^{n} and C t n C t n C_(t)^(n)C_{t}{ }^{n} there exists a quasiconformal mapping f : R n → R n f : R n → R n f:R^(n)rarrR^(n)f: R^{n} \rightarrow R^{n} which maps C s n C s n C_(s)^(n)C_{s}{ }^{n} onto C t n C t n C_(t)^(n)C_{t}{ }^{n}.
    定理。对于每个整数 n ⩾ 2 n ⩾ 2 n >= 2n \geqslant 2 和每一对这样的康托集 C s n C s n C_(s)^(n)C_{s}{ }^{n} 和 C t n C t n C_(t)^(n)C_{t}{ }^{n} ，存在一个拟共形映射 f : R n → R n f : R n → R n f:R^(n)rarrR^(n)f: R^{n} \rightarrow R^{n} 将 C s n C s n C_(s)^(n)C_{s}{ }^{n} 映射到 C t n C t n C_(t)^(n)C_{t}{ }^{n} 。
Proof. Let g i g i g_(i)g_{i} and F j , g i ′ F j , g i ′ F_(j),g_(i)^(')F_{j}, g_{i}{ }^{\prime} and F j ′ F j ′ F_(j)^(')F_{j}{ }^{\prime} denote respectively the similarity mappings and sets corresponding to the constructions for C s n , C t n C s n , C t n C_(s)^(n),C_(t)^(n)C_{s}{ }^{n}, C_{t}{ }^{n} given in § 4 § 4 §4\S 4§. Then it is not difficult to see that there exists a piecewise linear homeomorphism f 1 : R n → R n f 1 : R n → R n f_(1):R^(n)rarrR^(n)f_{1}: R^{n} \rightarrow R^{n} such that f 1 ( x ) = x f 1 ( x ) = x f_(1)(x)=xf_{1}(x)=x if x ∈ R n ∼ Q x ∈ R n ∼ Q x inR^(n)∼Qx \in R^{n} \sim Q and such that for each i i ii
证明。设 g i g i g_(i)g_{i} 、 F j , g i ′ F j , g i ′ F_(j),g_(i)^(')F_{j}, g_{i}{ }^{\prime} 和 F j ′ F j ′ F_(j)^(')F_{j}{ }^{\prime} 分别表示对应于 § 4 § 4 §4\S 4§ 中给出的 C s n , C t n C s n , C t n C_(s)^(n),C_(t)^(n)C_{s}{ }^{n}, C_{t}{ }^{n} 的相似映射和集合。那么不难看出存在一个分段线性同胚 f 1 : R n → R n f 1 : R n → R n f_(1):R^(n)rarrR^(n)f_{1}: R^{n} \rightarrow R^{n} ，使得当 x ∈ R n ∼ Q x ∈ R n ∼ Q x inR^(n)∼Qx \in R^{n} \sim Q 时 f 1 ( x ) = x f 1 ( x ) = x f_(1)(x)=xf_{1}(x)=x ，并且对于每个 i i ii 。
f 1 ( x ) = g i ′ ∘ g i − 1 ( x ) f 1 ( x ) = g i ′ ∘ g i − 1 ( x ) f_(1)(x)=g_(i)^(')@g_(i)^(-1)(x)f_{1}(x)=g_{i}^{\prime} \circ g_{i}^{-1}(x)
if x ∈ g i [ Q ] x ∈ g i [ Q ] x ing_(i)[Q]x \in g_{i}[Q]. Then f 1 f 1 f_(1)f_{1} is K K KK-quasiconformal for some K K KK and f 1 [ F 1 ] = F 1 ′ f 1 F 1 = F 1 ′ f_(1)[F_(1)]=F_(1)^(')f_{1}\left[F_{1}\right]=F_{1}{ }^{\prime}. Next define f 2 : R n → R n f 2 : R n → R n f_(2):R^(n)rarrR^(n)f_{2}: R^{n} \rightarrow R^{n} by setting f 2 ( x ) = f 1 ( x ) f 2 ( x ) = f 1 ( x ) f_(2)(x)=f_(1)(x)f_{2}(x)=f_{1}(x) if x ∈ R n ∼ F 1 x ∈ R n ∼ F 1 x inR^(n)∼F_(1)x \in R^{n} \sim F_{1} and
如果 x ∈ g i [ Q ] x ∈ g i [ Q ] x ing_(i)[Q]x \in g_{i}[Q] 。那么 f 1 f 1 f_(1)f_{1} 是 K K KK -拟共形的，对于某些 K K KK 和 f 1 [ F 1 ] = F 1 ′ f 1 F 1 = F 1 ′ f_(1)[F_(1)]=F_(1)^(')f_{1}\left[F_{1}\right]=F_{1}{ }^{\prime} 。接下来定义 f 2 : R n → R n f 2 : R n → R n f_(2):R^(n)rarrR^(n)f_{2}: R^{n} \rightarrow R^{n} ，通过设置 f 2 ( x ) = f 1 ( x ) f 2 ( x ) = f 1 ( x ) f_(2)(x)=f_(1)(x)f_{2}(x)=f_{1}(x) 如果 x ∈ R n ∼ F 1 x ∈ R n ∼ F 1 x inR^(n)∼F_(1)x \in R^{n} \sim F_{1} 。
f 2 ( x ) = g i ′ ∘ f 1 ∘ g i − 1 ( x ) f 2 ( x ) = g i ′ ∘ f 1 ∘ g i − 1 ( x ) f_(2)(x)=g_(i)^(')@f_(1)@g_(i)^(-1)(x)f_{2}(x)=g_{i}^{\prime} \circ f_{1} \circ g_{i}^{-1}(x)
if x ∈ g i [ Q ] x ∈ g i [ Q ] x ing_(i)[Q]x \in g_{i}[Q]. Then f 2 f 2 f_(2)f_{2} is a piecewise linear K K KK-quasiconformal mapping, f 2 [ F 2 ] = F 2 ′ f 2 F 2 = F 2 ′ f_(2)[F_(2)]=F_(2)^(')f_{2}\left[F_{2}\right]=F_{2}{ }^{\prime}, and for each i i ii and j j jj
如果 x ∈ g i [ Q ] x ∈ g i [ Q ] x ing_(i)[Q]x \in g_{i}[Q] 。那么 f 2 f 2 f_(2)f_{2} 是一个分段线性的 K K KK -拟共形映射 f 2 [ F 2 ] = F 2 ′ f 2 F 2 = F 2 ′ f_(2)[F_(2)]=F_(2)^(')f_{2}\left[F_{2}\right]=F_{2}{ }^{\prime} ，并且对于每个 i i ii 和 j j jj
f 2 ( x ) = g i ′ ∘ g j ′ ∘ g j − 1 ∘ g i − 1 ( x ) f 2 ( x ) = g i ′ ∘ g j ′ ∘ g j − 1 ∘ g i − 1 ( x ) f_(2)(x)=g_(i)^(')@g_(j)^(')@g_(j)^(-1)@g_(i)^(-1)(x)f_{2}(x)=g_{i}^{\prime} \circ g_{j}^{\prime} \circ g_{j}^{-1} \circ g_{i}^{-1}(x)
if x ∈ g i ∘ g j [ Q ] x ∈ g i ∘ g j [ Q ] x ing_(i)@g_(j)[Q]x \in g_{i} \circ g_{j}[Q]. Continuing in this way, we obtain a sequence of piecewise linear K K KK-quasiconformal mappings f j : R n → R n f j : R n → R n f_(j):R^(n)rarrR^(n)f_{j}: R^{n} \rightarrow R^{n} such that f j + 1 ( x ) = f j ( x ) f j + 1 ( x ) = f j ( x ) f_(j+1)(x)=f_(j)(x)f_{j+1}(x)=f_{j}(x) in R n ∼ F j R n ∼ F j R^(n)∼F_(j)R^{n} \sim F_{j} and f j [ F j ] = F j ′ f j F j = F j ′ f_(j)[F_(j)]=F_(j)^(')f_{j}\left[F_{j}\right]=F_{j}{ }^{\prime}. This sequence converges to a K K KK-quasiconformal mapping f : R n → R n f : R n → R n f:R^(n)rarrR^(n)f: R^{n} \rightarrow R^{n} which maps F j F j F_(j)F_{j} onto F j ′ F j ′ F_(j)^(')F_{j}{ }^{\prime} for each j j jj. Hence f f ff maps C s n C s n C_(s)^(n)C_{s}{ }^{n} onto C t n C t n C_(t)^(n)C_{t}{ }^{n}.
如果 x ∈ g i ∘ g j [ Q ] x ∈ g i ∘ g j [ Q ] x ing_(i)@g_(j)[Q]x \in g_{i} \circ g_{j}[Q] 。继续这种方式，我们得到一个分段的线性 K K KK -拟共形映射 f j : R n → R n f j : R n → R n f_(j):R^(n)rarrR^(n)f_{j}: R^{n} \rightarrow R^{n} 序列，使得 f j + 1 ( x ) = f j ( x ) f j + 1 ( x ) = f j ( x ) f_(j+1)(x)=f_(j)(x)f_{j+1}(x)=f_{j}(x) 在 R n ∼ F j R n ∼ F j R^(n)∼F_(j)R^{n} \sim F_{j} 中，并且 f j [ F j ] = F j ′ f j F j = F j ′ f_(j)[F_(j)]=F_(j)^(')f_{j}\left[F_{j}\right]=F_{j}{ }^{\prime} 。这个序列收敛到一个 K K KK -拟共形映射 f : R n → R n f : R n → R n f:R^(n)rarrR^(n)f: R^{n} \rightarrow R^{n} ，它将 F j F j F_(j)F_{j} 映射到 F j ′ F j ′ F_(j)^(')F_{j}{ }^{\prime} 对每个 j j jj 。因此 f f ff 将 C s n C s n C_(s)^(n)C_{s}{ }^{n} 映射到 C t n C t n C_(t)^(n)C_{t}{ }^{n} 。
6. Corollary. For each integer n ⩾ 2 n ⩾ 2 n >= 2n \geqslant 2 and each pair of numbers α , β ∈ ( 0 , n ) α , β ∈ ( 0 , n ) alpha,beta in(0,n)\alpha, \beta \in(0, n), there exists a quasiconformal mapping f : R n → R n f : R n → R n f:R^(n)rarrR^(n)f: R^{n} \rightarrow R^{n} and a compact set A ⊂ R n A ⊂ R n A subR^(n)A \subset R^{n} such that
6. 推论。对于每个整数 n ⩾ 2 n ⩾ 2 n >= 2n \geqslant 2 和每一对数 α , β ∈ ( 0 , n ) α , β ∈ ( 0 , n ) alpha,beta in(0,n)\alpha, \beta \in(0, n) ，存在一个拟共形映射 f : R n → R n f : R n → R n f:R^(n)rarrR^(n)f: R^{n} \rightarrow R^{n} 和一个紧致集 A ⊂ R n A ⊂ R n A subR^(n)A \subset R^{n} 使得
H − dim A = α , H − dim f [ A ] = β . H − dim A = α , H − dim f [ A ] = β . H-dim A=alpha,H-dim f[A]=beta.\mathrm{H}-\operatorname{dim} A=\alpha, \mathrm{H}-\operatorname{dim} f[A]=\beta .
Proof. By (7) and (8) we can choose s , t ∈ ( 0 , 1 2 ) s , t ∈ 0 , 1 2 s,t in(0,(1)/(2))s, t \in\left(0, \frac{1}{2}\right) so that for any of the corresponding Cantor sets C s n , C t n C s n , C t n C_(s)^(n),C_(t)^(n)C_{s}{ }^{n}, C_{t}{ }^{n},
证明。由 (7) 和 (8) 我们可以选择 s , t ∈ ( 0 , 1 2 ) s , t ∈ 0 , 1 2 s,t in(0,(1)/(2))s, t \in\left(0, \frac{1}{2}\right) 使得对于相应的任何康托集 C s n , C t n C s n , C t n C_(s)^(n),C_(t)^(n)C_{s}{ }^{n}, C_{t}{ }^{n} ，
H − dim C s n = α , H − dim C t n = β . H − dim C s n = α , H − dim C t n = β . H-dimC_(s)^(n)=alpha,H-dimC_(t)^(n)=beta.\mathrm{H}-\operatorname{dim} C_{s}{ }^{n}=\alpha, \mathrm{H}-\operatorname{dim} C_{t}{ }^{n}=\beta .
Theorem 5 then yields a quasiconformal mapping f : R n → R n f : R n → R n f:R^(n)rarrR^(n)f: R^{n} \rightarrow R^{n} which maps C s n C s n C_(s)^(n)C_{s}{ }^{n} onto C t n C t n C_(t)^(n)C_{t}{ }^{n}, and (9) follows with A = C s n A = C s n A=C_(s)^(n)A=C_{s}{ }^{n}.
定理 5 就给出了一个拟共形映射 f : R n → R n f : R n → R n f:R^(n)rarrR^(n)f: R^{n} \rightarrow R^{n} 将 C s n C s n C_(s)^(n)C_{s}{ }^{n} 映射到 C t n C t n C_(t)^(n)C_{t}{ }^{n} ，并且 (9) 随着 A = C s n A = C s n A=C_(s)^(n)A=C_{s}{ }^{n} 而成立。
7. Remark. The above proof shows that for each α ∈ ( 0 , n ) α ∈ ( 0 , n ) alpha in(0,n)\alpha \in(0, n) there exists a set A ⊂ R n A ⊂ R n A subR^(n)A \subset R^{n} with H- dim A = α dim A = α dim A=alpha\operatorname{dim} A=\alpha such that
7. 注记。上述证明表明，对于每个 α ∈ ( 0 , n ) α ∈ ( 0 , n ) alpha in(0,n)\alpha \in(0, n) 都存在一个具有 H- dim A = α dim A = α dim A=alpha\operatorname{dim} A=\alpha 的集 A ⊂ R n A ⊂ R n A subR^(n)A \subset R^{n} 使得
inf f H − dim f [ A ] = 0 , sup f H − dim f [ A ] = n , inf f   H − dim f [ A ] = 0 , sup f   H − dim f [ A ] = n , i n f_(f)H-dim f[A]=0,s u p_(f)H-dim f[A]=n,\inf _{f} \mathrm{H}-\operatorname{dim} f[A]=0, \sup _{f} \mathrm{H}-\operatorname{dim} f[A]=n,
where the infimum and supremum are taken over all quasiconformal mappings f : D → D ′ f : D → D ′ f:D rarrD^(')f: D \rightarrow D^{\prime} with A ⊂ D A ⊂ D A sub DA \subset D. We consider next what can be said if we take the infimum and supremum in (10) over the subclass of mappings f : D → D ′ f : D → D ′ f:D rarrD^(')f: D \rightarrow D^{\prime} which are K K KK quasiconformal for some fixed K K KK.
在所有满足条件的拟共形映射 f : D → D ′ f : D → D ′ f:D rarrD^(')f: D \rightarrow D^{\prime} 上取下确界和上确界。接下来，我们考虑如果我们在(10)中对满足某些固定 K K KK 的 K K KK 的子类映射取下确界和上确界，可以得出什么结论。
8. Theorem. If f : D → D ′ f : D → D ′ f:D rarrD^(')f: D \rightarrow D^{\prime} is an n-dimensional K K KK-quasiconformal mapping and if A ⊂ D A ⊂ D A sub DA \subset D with H − dim A ⩾ α > 0 H − dim A ⩾ α > 0 H-dim A >= alpha > 0\mathrm{H}-\operatorname{dim} A \geqslant \alpha>0, then H − dim f [ A ] ⩾ β > 0 H − dim f [ A ] ⩾ β > 0 H-dim f[A] >= beta > 0\mathrm{H}-\operatorname{dim} f[A] \geqslant \beta>0, where
8. 定理。如果 f : D → D ′ f : D → D ′ f:D rarrD^(')f: D \rightarrow D^{\prime} 是一个 n 维的 K K KK -拟共形映射，并且如果 A ⊂ D A ⊂ D A sub DA \subset D 满足 H − dim A ⩾ α > 0 H − dim A ⩾ α > 0 H-dim A >= alpha > 0\mathrm{H}-\operatorname{dim} A \geqslant \alpha>0 ，那么 H − dim f [ A ] ⩾ β > 0 H − dim f [ A ] ⩾ β > 0 H-dim f[A] >= beta > 0\mathrm{H}-\operatorname{dim} f[A] \geqslant \beta>0 ，其中
β = α K 1 / ( 1 − n ) ⩾ α / K . β = α K 1 / ( 1 − n ) ⩾ α / K . beta=alphaK^(1//(1-n)) >= alpha//K.\beta=\alpha K^{1 /(1-n)} \geqslant \alpha / K .
Proof. Since A A AA is the countable union of sets with compact closure in D D DD, we may assume that A A AA is contained in a compact subset of D D DD. Then since f − 1 f − 1 f^(-1)f^{-1} is locally
证明。由于 A A AA 是 D D DD 中具有紧闭集的集合的可数并，我们可以假设 A A AA 包含在一个 D D DD 的紧子集内。然后由于 f − 1 f − 1 f^(-1)f^{-1} 在局部
  1. Received 4 November, 1971.
    收到于1971年11月4日。
    † † †\dagger This research was supported in part by the National Science Foundation, Contracts GP 7041X and GP 28115.
    † † †\dagger 这项研究部分得到了国家科学基金会的支持，合同 GP 7041X 和 GP 28115。
    [J. London Math. Soc. (2), 6 (1973), 504-512]