On Typical Compact Submanifolds of the Euclidean Space
It is shown that typical compact submanifolds of R n are nowhere differentiable with integer box dimensions.
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| author | Mirzaie, R. Мірзай, Р. |
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| description | It is shown that typical compact submanifolds of R n are nowhere differentiable with integer box dimensions. |
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UDC 517.91
R. Mirzaie (Imam Khomeini Int. Univ., Qazvin, Iran)
ON TYPICAL COMPACT SUBMANIFOLDS OF EUCLIDEAN SPACE
ПРО ТИПОВI КОМПАКТНI ПIДМНОГОВИДИ ЕВКЛIДОВОГО ПРОСТОРУ
We show that typical compact submanifolds of Rn are nowhere differentiable with integer box dimensions.
Показано, що типовi компактнi пiдмноговиди простору Rn нiде не диференцiйовнi при цiлих розмiрностях Мiн-
ковського.
1. Introduction. A subset Y of a topological space X is called to be comeagre, if there is a countable
collection {Wi} of open and dense subsets of X such that
⋂
i
Wi ⊂ Y. Complement of a comeagre
subset is called a meagre subset. A meagre subset can be considered as a countable union of nowhere
dense subsets and they are negligible in some sense. So, we say that some property holds for typical
elements of X, if it holds on a comeagre subset. Let X be a metric space and C(X) be the set of all
compact subsets of X. The Hausdorff metric dH is defined on C(X) by
dH(E,F ) = max
{
sup
x∈E
inf
y∈F
d(x, y), sup
y∈F
inf
x∈E
d(x, y)
}
.
We will denote by K(X) the set of all connected compact subsets of X. Study of properties of
typical elements of X, C(X) and K(X) is a classic and interesting part of mathematics. It is proved
in [8] that typical elements of C(X) have zero Hausdorff dimensions. A well known theorem due
to Banach states that typical elements of the set of all real valued continuous functions defined on
[0, 1] are nowhere differentiable. One can see many other interesting results in [2, 3, 5, 8, 10, 11].
It is proved that a typical element of K(Rn) consists of a number of slightly blurred line segments.
Typical elements of the set of graphs of all curves in Rn, starting at a fixed point, have Hausdorff
dimension 1 (see [5]). It is proved in [3] that if M is a compact differentiable manifold with boundary,
imbedded in Rn, and S is the set of all deformations of the boundary of M, then typical elements of
S are nowhere differentiable with integer box dimensions. We show in the present paper that similar
results are true on a more general case, for the set of all compact topological submanifolds of Rn.
Our main results are Theorems 3.1 and 3.2.
2. Preliminaries. The following notations will be used in the proofs:
(1) Ωn = {M : M is a compact topological submanifold of Rn}.
(2) DΩn = {M ∈ Ωn : M is differentiable}.
(3) NDΩn = {M ∈ Ωn : M is nowhere differentiable}.
(4) Bε = {x ∈ R : |x| < ε}, Bk
(ε) = Bε × . . .×Bε (k times).
(5) I = [−1, 1], Ik = I × I × . . .× I (k times).
(6) If M ∈ DΩn and U is an open subset of Rn, then
C(M,U) = {f : M → U ; f is countinuos}.
(7) D(M,U) = {f ∈ C(M,U) : f is differentible}.
c© R. MIRZAIE, 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7 1009
1010 R. MIRZAIE
(8) If M ∈ DΩn, then we will denote by ND(M,U) the set of all nowhere differentiable
members of C(M,U).
Let M be a bounded subset of Rn. We denote by dim(M) the topological dimension of M. For
each number ε > 0 put
]ε(M) = sup{card {Z} : Z ⊂M and for each x, y ∈ Z, |x− y| > ε}.
The upper and lower box dimensions of M are defined by
dimB(M) = lim sup
ε→0
log ]ε(M)
− log ε
,
dimB(M) = lim inf
ε→0
log ]ε(M)
− log ε
.
If dimB(M) = dimB(M), then dimB(M) = limε→0
log ]ε(M)
− log ε
is the box dimension of M.
Notice 2.1. If M is a differentiable submanifold of Rn and dim(M) = m, then
(1) dimB(M) = dim(M) = m.
(2) If g : M → Rn is a differentiable map and Mg = g(M), then
dimB(Mg) = dim(Mg) ∈ {0, 1, . . . ,m}.
If M is a compact manifold, then C(M,Rn) endowed with the following metric d is a complete
metric space
d(f, g) = max
x∈M
|f(x)− g(x)|.
The following theorem due to Banach is well known.
Theorem 2.1 [1]. Typical elements of C(I,R) are nowhere differentiable.
It is easy to show that Banach’s theorem is also true if we replace C(I,R) by C(I,Bε).
The following lemma is a generalization of Banach’s theorem.
Lemma 2.1. If M is a differentiable compact manifold and ε > 0, then typical elements of
C(M,Bk
ε ) with the above metric d, are nowhere differentiable.
Proof. We give the proof in the following steps.
Step 1. For each k ∈ N, ND(Ik, Bε) is a comeagre subset of C(Ik, Bε).
Proof. The claim is true for k = 1 (Banach’s theorem). Suppose that the claim is true for each
natural number m, m ≤ k. We show that it is true for k + 1. Let h ∈ C(Ik+1, Bε) and t ∈ I. Put
ht : I
k → Bε,
ht(x) = h(x, t)
and let
Γ = {h ∈ C(Ik+1, Bε) : ∀t ∈ I, ht is nowhere differentiable}.
We show that Γ is a comeagre subset of C(Ik+1, Bε).
Consider the set
∏
t∈I
C(Ik, Bε)t, C(Ik, Bε)t = C(Ik, Bε) and put
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
ON TYPICAL COMPACT SUBMANIFOLDS OF EUCLIDEAN SPACE 1011
σ : C(Ik+1, Bε)→
∏
t∈I
C(Ik, Bε)t,
σ(h) =
∏
t
(ht),
W (f, k, δ) = {g ∈ C(Ik, Bε) : d(g, f) < δ}, δ > 0, f ∈ C(Ik, Bε).
Let O be an open subset of C(Ik, Bε) and put U =
∏
tOt, Ot = O. If h ∈ C(Ik+1, Bε), then the
function α : I → O defined by α(t) = ht is continuous. Due to compactness of I, we can find a
number δ > 0 such that for all t ∈ I, W (ht, k, δ) ⊂ O. Then W (h, k+ 1, δ) ⊂ σ−1(U). This means
that σ−1(U) is open in C(Ik+1, Bε).
By assumption, ND(Ik, Bε) is a comeagre subset of C(Ik, Bε). So, there is a countable collec-
tion {Om : m ∈ N} of open and dense subsets of C(Ik, Bε) such that⋂
m∈N
Om ⊂ ND(Ik, Bε).
Let
Um =
∏
t
(Om)t, (Om)t = Om.
σ−1(Um) is open in C(Ik+1, Bε) and we have
⋂
m∈N
σ−1(Um) ⊂ Γ. Also, it is not hard to show
that for each m ∈ N, σ−1(Um) is a dense subset of C(Ik+1, Bε). Now, from the fact that Γ ⊂
⊂ ND(Ik+1, Bε), we get the result.
Step 2. ND(M,Bε) is comeagre in C(M,Bε).
Proof. Let k = dimM and for each point p ∈ M consider a chart (O,ψ) around p such
that Ik ⊂ ψ(O). Since M is compact then there is a finite collection of this kind of charts, say
{(O1, ψ1), . . . , (Ol, ψl)}, such that M ⊂ ψ−1
1 (Ik) ∪ . . . ∪ ψ−1
l (Ik). Put Ui = ψ−1
i (Ik), 1 ≤ i ≤ l,
and for each h ∈ C(M,Bε) denote by hi the restriction of h on Ui, and consider the following
function:
ϕi : C(M,Bε)→ C(Ui, Bε), ϕi(h) = hi.
Since ψ(Ui) = Ik then we get from Step 1, that ND(Ui, Bε) is a comeagre subset of C(Ui, Bε). So
there is a countable collection {W i
m : m ∈ N} of open and dense subsets of C(Ui, Bε) such that⋂
m
W i
m ⊂ ND(Ui, Bε).
We show that for each i,m ∈ N, ϕ−1
i (W i
m) is a dense subset of C(M,Bε). Suppose h ∈ C(M,Bε)
and let δ > 0. Since W i
m is dense in C(Ui, Bε), then there is a function f ∈W i
m such that
d(hi, f) <
δ
2
. (2.1)
Let f̂ : M → Bε be a continuous extension of f on M. Since h and f̂ are continuous, then by (2.1),
there is an open subset B of M such that Ui ⊂ B and
x ∈ B ⇒ d(h(x), f̂(x)) < δ. (2.2)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
1012 R. MIRZAIE
Now let η : M → [0, 1] be a continuous function such that
η(x) = 1 for x ∈ Ui and η(x) = 0 for x ∈M −B.
Put
τ(x) = h(x) + η(x)(f̂(x)− h(x)). (2.3)
Then
|h(x)− τ(x)| = |η(x)|f̂(x)− h(x)| < δ.
Since the image of h is compact and included in Bε then for sufficiently small δ, the image of τ
will be included in Bε, so τ ∈ C(M,Bε). If x ∈ Ui, then τ(x) = f(x), so ϕi(τ) = f. Thus
τ ∈ ϕ−1
i (W i
m). This means that ϕ−1
i (W i
m) is dense in C(M,Bε). It is easy to show that⋂
m∈N
⋂
1≤i≤l
ϕ−1
i (W i
m) ⊂ ND(M,Bε).
Therefore, ND(M,Bε) is a comeagre subset of C(M,Bε).
Step 3. Proof of the lemma.
For each h ∈ C(M,Bk
ε ) we have h = (h1, . . . , hk) such that hi ∈ C(M,Bε). Consider the map
ψ : C(M,Bk
ε )→ C(M,Bε)× . . .× C(M,Bε) (k times),
ψ(h) = (h1, . . . , kk),
ψ is a homeomorphism and
ψ−1[ND(M,Bε)× . . .×ND(M,Bε)] ⊂ ND(M,Bk
ε ). (2.4)
Since by Step 2, ND(M,Bε) is comeagre in C(M,Bε), then ND(M,Bε) × . . . ×ND(M,Bε) is
comeagre in C(M,Bε) × . . . × C(M,Bε). Thus ψ−1[ND(M,Bε) × . . . × ND(M,Bε)] must be
comeagre in C(M,Bk
ε ). Now, we get the result by (2.4).
3. Main results.
Theorem 3.1. Typical elements of the set of compact submanifolds of Rn are nowhere differ-
entiable.
Proof. Let M be a differentiable compact submanifold of Rn. If k = n − dimM and p ∈ M,
then Rk can be considered as the set of all vectors perpendicular to M at p. For each v ∈ Rk denote
by vp the corresponding vector in TpM⊥. Since M is compact then there is an ε > 0 such that the
following map ψ, is a diffeomorphism from M ×Bk
ε onto an open neighborhood of M in Rn:
ψ : M ×Bk
ε → Rn, ψ(p, v) = p+ vp.
For each g ∈ C(M,Bk
ε ) let Mg = {(ψ(x, g(x)) : x ∈M)} and put
λ(M) = {Mg : g ∈ C(M,Bε)},
ND(λ(M)) = {Mg ∈ λ(M) : g is nowhere differentiable}.
Consider the following metric d on λ(M):
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
ON TYPICAL COMPACT SUBMANIFOLDS OF EUCLIDEAN SPACE 1013
d(Mg,Mh) = sup
x∈M
|g(x)− h(x)|.
By using of Lemma 2.1, we get that typical elements of λ(M) with the metric d are nowhere
differentiable. Then it is easy to show that typical elements of λ(M) with the Hausdorff metric are
also nowhere differentiable. Now consider the following subspace of C(Rn):
Λ(Rn) =
⋃
M∈DΩn
λ(M).
We show that typical elements of Λ(Rn) (with the Hausdorff metric) are nowhere differentiable.
Since for each differentiable submanifold M of Rn, typical elements of λ(M) are nowhere
differentiable then there is a collection {O(M,i) : i ∈ N} of open and dense subsets of λ(M) such
that ⋂
i∈N
O(M,i) ⊂ ND(λ(M)). (3.1)
Since λ(M) is a subspace of Ωn, for each i ∈ N there is a countable collection {U(M,i,j) : j ∈ N}
of open subsets of Ωn such that O(M,i) = U(M,i,j)
⋂
λ(M) and
sup {dH(Mg,Mh) ∈ O(M,i) × U(M,i,j)} <
1
j
. (3.2)
Now put
W(M,i,j) = U(M,i,j) − {x : x is a boundary point of O(M,i) in Ωn}. (3.3)
We get from (3.2) and (3.3) that ⋂
j
W(M,i,j) = O(M,i). (3.4)
Let
Wi,j =
⋃
M∈DΩn
W(M,i,j).
If ND(ΛRn) = {M ∈ ΛRn : M is nowhere differentiable}, then by (3.1) and (3.4)⋂
i,j∈N
Wi,j ⊂ ND(Λ(Rn)).
Since for each i, O(M,i) is dense in λ(M), then for each i, j, Wi,j ∩ Λ(Rn) is dense in Λ(Rn).
Also the set of differentiable submanifolds of Rn is dense in Ωn, so Wi,j is dense in Ωn. Therefore,
ND(Λ(Rn)) is a comeagre subset of Λ(Rn). Now we get the result from the fact that ND(Λ(Rn)) ⊂
⊂ NDΩn.
Theorem 3.2. Typical elements of the set of compact submanifolds of Rn have integer box
dimensions.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
1014 R. MIRZAIE
Proof. Suppose M,N ∈ Ωn and dH(M,N) < ε. Let O1, . . . , O]ε(N) be balls with radius ε such
that
N ⊂
⋃
i
Oi.
For each 1 ≤ i ≤ ]ε(N), let Ôi be the ball of radius 2ε with the same center as Oi. Each Ôi can be
covered by 4n balls with radius ε. Thus
]ε(M) ≤ 4n]ε(N).
In a similar way, we can show that ]ε(N) ≤ 4n]ε(M). Then
4−n]ε(M) ≤ ]ε(N) ≤ 4n]ε(M).
Therefore,
−n log 4 + log ]ε(M)
− log ε
≤ log ]ε(N)
− log ε
≤ n log 4 + log ]ε(M)
− log ε
.
If M is differentiable then dimB(M) is an integer ≤ n. Thus limε→∞
log ]ε(M)
− log ε
= dimM ∈
∈ {0, 1, . . . , n}. Then for each k ∈ N there is an open neighborhood Uk,M of M in Ωn such that for
each N ∈ Uk,M
dimM − 1
k
≤ log ]ε(N)
− log ε
≤ dimM +
1
k
.
Put Wk =
⋃
M∈DΩn
Uk,M . Since DΩn is dense in Ωn then for any k ∈ N, Wk is dense in Ωn. Now
put
W =
⋂
k
Wk.
W is comeagre in Ωn and for each N ∈W, dimB N is an integer number.
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Received 28.05.12,
after revision — 14.11.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 7
|
| id | umjimathkievua-article-2486 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:24:20Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/54/d0f39e20d0a655dc8e46e96415665554.pdf |
| spelling | umjimathkievua-article-24862020-03-18T19:16:28Z On Typical Compact Submanifolds of the Euclidean Space Про типові компактні підмноговиди евклідового простору Mirzaie, R. Мірзай, Р. It is shown that typical compact submanifolds of R n are nowhere differentiable with integer box dimensions. Показано, що типові компактні підмноговиди простору R n нідє не диференційовні при цілих розмірностях Мінковського. Institute of Mathematics, NAS of Ukraine 2013-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2486 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 7 (2013); 1009–1014 Український математичний журнал; Том 65 № 7 (2013); 1009–1014 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2486/1738 https://umj.imath.kiev.ua/index.php/umj/article/view/2486/1739 Copyright (c) 2013 Mirzaie R. |
| spellingShingle | Mirzaie, R. Мірзай, Р. On Typical Compact Submanifolds of the Euclidean Space |
| title | On Typical Compact Submanifolds of the Euclidean Space |
| title_alt | Про типові компактні підмноговиди евклідового простору |
| title_full | On Typical Compact Submanifolds of the Euclidean Space |
| title_fullStr | On Typical Compact Submanifolds of the Euclidean Space |
| title_full_unstemmed | On Typical Compact Submanifolds of the Euclidean Space |
| title_short | On Typical Compact Submanifolds of the Euclidean Space |
| title_sort | on typical compact submanifolds of the euclidean space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2486 |
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