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\listok{3}{Geometry 3: Hausdorff dimension}

{\scriptsize
{\bf Rules:} You may choose to solve only 
``hard'' exercises (marked with !, * and **) 
or ``ordinary'' ones (marked with ! or unmarked),
or both, if you want to have extra problems.
To have a perfect score, a student must obtain
(in average) a score of 10 points per week.
It's up to you to ignore handouts entirely,
because passing tests in class and having
good scores at final exams could compensate
(at least, partially) for the points obtained 
by grading handouts.

Solutions for the problems are to be explained to the 
examiners orally in the class and marked in the score sheet. 
It's better to have a written version of your solution with 
you. It's OK to share your solutions with other students, and use
books, Google search and Wikipedia, we encourage it.
The first score sheet will be distributed
February 11-th. 

If you have got credit for 2/3 of ordinary problems
or 2/3 of ``hard'' problems, you receive  
$6t$ points, where $t$ is a number depending on the
date when it is done. Passing all ``hard'' 
or all ``ordinary'' problems
(except at most 2) brings you $10t$ points.
Solving of ``**'' (extra hard) problems is not
obligatory, but each such problem gives you a credit
for 2 ``*'' or ``!'' problems in the ``hard'' set.

The first 3 weeks after giving a handout, $t=1.5$,
between 21 and 35 days, $t=1$, and afterwards, $t=0.7$.
The scores are not cumulative, only the
best score for each handout counts.

Please keep your score sheets until the final
evaluation is given.

The original English translation of this handout
was done by Sasha Anan$'$in (UNICAMP) in 2010.
}

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\subsection{Hausdorff dimension and measure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition Let $M$ be a metric space. The {\bf diameter}
$\diam M\in[0,\infty]$ is the number $\sup\limits_{x,y\in M}d(x,y)$.
\ed

\definition In a metric space, {\bf a ball} of
radius $\varepsilon$ centered at $x$ is defined as the set of all
points $y$ satisfying $d(x,y)<\varepsilon$.
\ed

\exercise
Describe all possible values of the diameter of the
ball of radius $\varepsilon$ in a metric space.
\ez

\exercise Let $M$ be a metric space and let $\varepsilon>0$.
Show that $M$ admits a cover by balls of diameter $\le\varepsilon$.
\ez

\definition Let $\{S_i\}$ be a cover of a metric space $M$
by balls of radius $r$ with $r<\varepsilon$. Define
$\mu_{d,\varepsilon}\in[0,\infty]$ as
$$\mu_{d,\varepsilon}M:=\inf_{\{S_i\}}\sum_i(\diam S_i)^d,$$
where the infimum is taken over all such covers. The
limit
$$\mu_dM:=\sup\lim_{\varepsilon\to0}\mu_{d,\varepsilon}M$$
is called {\bf $d$-dimensional Hausdorff measure} of $M$.
\ed

\exercise Consider $M=\Bbb R^n$ with a metric  given by
the norm \[ \big|(x_1,\dots,x_n)\big|:=\max|x_i|.\] Show that the
$n$-dimensional Hausdorff measure of a polyhedron equals its volume (in
the usual sense).
\ez

\exercise Consider the metric on $M=\Bbb R^n$ given
by the norm $\big|(x_1,\dots,x_n)\big|:=\sum|x_i|$. 
\enum
\ite Prove that the
$n$-dimensional Hausdorff measure of a polyhedron is proportional to
its volume. 
\ite[*] Calculate the coefficient of proportionality.
\ee
\ez

\exercise
Consider  $M=\Bbb R^n$ with the usual (Euclidean)
metric. 
\enum
\ite Show that the
$n$-dimensional Hausdorff measure of a polyhedron is proportional to
its volume. 
\ite[*] Calculate the coefficient of proportionality.
\ee
\ez

\definition A map $f:M\to N$ of metric spaces is called
Lipschitz with constant $C>0$ if
$d(x,y)\ge C\cdot d\big(f(x),f(y)\big)$ for all $x,y\in M$. A map is
called bi-Lipschitz if it is bijective and the inverse map is also
Lipschitz (with some constant).
\ed

\exercise
Show that every Lipschitz map is continuous.
\ez

\exercise[*] Construct an example of a continuous map of
metric spaces that is not Lipschitz.
\ez


\exercise Let $d_1,d_2$ be two norms on a vector space $V$.
Denote the corresponding metrics by the same letters. Prove that the
identity map $\Id_V:(V,d_1)\arrow(V,d_2)$ is Lipschitz if and only if the
unit ball $B_1(r,d_1)$ is bounded in the metric $d_2$.
\ez

\exercise[*]
Let $M=\Bbb R^n$ and let $d_1,d_2$ be some
norms on $M$. Show that $\Id_M:(M,d_1)\to(M,d_2)$ is bi-Lipschitz.
\ez

\exercise[!] Let $U\subset\Bbb R^n$ be a bounded open subset
and $\Phi:U\to\Bbb R^n$ a smooth map which
can be smoothly extended to the
boundary $\partial U$. Prove that $\Phi$ is Lipschitz.
\ez


\exercise[!] Let $M\stackrel f\arrow N$ be a
Lipschitz map of metric spaces with constant $C$. Show that
$\mu_dM\ge C^d\mu_df(M)$, where $\mu_d$ is $d$-dimensional
Hausdorff measure on $M$.
\ez

\exercise[!] Suppose that $\mu_dM<\infty$. Show that
$\mu_{d'}M=0$ for every $d'>d$.
\ez

\hint Deduce from $\diam S_i<\varepsilon$ the inequality
\begin{equation}\label{_long_Hausd_Equation_}
\mu_{d',\varepsilon}M=\inf\limits_{\{S_i\}}\sum\limits_i
(\diam S_i)^{d'}\le\varepsilon^{d'-d}\inf\limits_{\{S_i\}}\sum\limits_i
(\diam S_i)^d=\varepsilon^{d'-d}\mu_{d,\varepsilon}M
\end{equation}
and pass to the limit $\varepsilon\to0$.
\eh

\exercise[!] Suppose that $\mu_{d'}M=\infty$. Show that
$\mu_dM=\infty$ for every $d<d'$.
\ez

\hint Use the inequality \eqref{_long_Hausd_Equation_} 
and pass to the limit $\varepsilon\to0$.
\eh

\definition Let $M$ be a metric space. The {\bf Hausdorff
dimension} $\dim_HM\in[0,\infty]$ is the supremum of all $d$ such that
$\mu_dM=\infty$.
\ed

\exercise 
Find the Hausdorff dimension of a finite set.
\ez

\exercise Let $f:M\to N$ be a Lipschitz map. Show that $f$
does not increase the Hausdorff dimension: $\dim_HM\ge\dim_Hf(M)$.
\ez

\exercise Show that every bi-Lipschitz map preserve Hausdorff
dimension (``Hausdorff dimension is a bi-Lipschitz invariant'').
\ez

\exercise[*] Find the Hausdorff dimension of the Cantor set
$K\subset[0,1]$, obtained as a set of all real numbers
without a number 1 in their ternary expansion.
\ez

\definition A subset $Z\subset\Bbb R^n$ has {\bf measure
zero} if for every $\varepsilon>0$ there exists a countable cover of
$Z$ by balls $U_i$ such that $\sum_i\Vol U_i<\varepsilon$.
\ed

\exercise Show that the countable union of subsets of zero
measure has measure zero.
\ez

\exercise Show that the image of a subset of zero measure
under a Lipschitz map $\Bbb R^n\to\Bbb R^n$ has measure zero.
\ez

\exercise[!] Show that the image of a subset of zero measure
under a smooth map $\Bbb R^n\to\Bbb R^n$ has measure zero.
\ez

\hint 
Prove that a smooth map is Lipschitz, and 
use the previous exercise.
\eh

\exercise[*]
Construct an example of a continuous map from
$\Bbb R^n$ to $\Bbb R^n$ that sends a subset of zero measure to a
subset of nonzero measure.
\ez

\exercise[!] Let $M\subset\Bbb R^d$ be a subset such that
$\dim_HM<d$. Show that $M$ has measure zero.
\ez

\definition Let $M$ be a smooth manifold with a countable atlas
$\{U_i,\ \varphi_i:U_i\to\Bbb R^n\}$. A subset $Z\subset M$ has {\bf
measure zero} if the image $\varphi(Z\cap U_i)$ has measure zero in
$\Bbb R^n$ for every $i$.
\ed

\exercise Show that this definition does not depend on the
choice of an atlas on $M$.
\ez

\exercise
Let $M\stackrel f\longrightarrow\Bbb R^n$ be a
smooth map of manifolds and let $M$ be a union of compact subsets. Show
that $\dim_Hf(M)\le\dim M$.
\ez

\hint Show first that $f$ is Lipschitz on compact subsets. Then
use the fact that Lipschitz maps satisfy $\dim_Hf(M)\le\dim M$.
\eh

\exercise[!] Let $M\stackrel f\longrightarrow N$ be a
smooth map of manifolds such that $\dim M<\dim N$. Show that the image
of $M$ has measure zero.
\ez

\hint Use the previous exercise.
\eh

\remark This theorem is a special case of {\bf Sard's lemma} that
claims that the set of critical values of a smooth map has measure
zero.
\er

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Whitney's theorem (with a bound on dimension}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\definition The {\bf Klein bottle} is the quotient of
the two-dimensional torus $T^2:=\Bbb S^1\times\Bbb S^1$ by the action
of the group $\Bbb Z/2\Bbb Z$ mapping $(t_1,t_2)$ to $(t_1+\pi,-t_2)$.
\ed

\exercise Show that this action is free, and the
quotient is a manifold.
\ez


\exercise
Let $M\stackrel f \arrow N$ be a smooth map of manifolds,
$f(x)=y$, and $U\ni x$, $V\ni y$ charts, equipped
with the embeddings $U \hookrightarrow \R^m$,
$V \hookrightarrow \R^n$. Choose $U$ and $V$ in such
a way that $f(U)\subset V$, and consider $f\restrict U$
as a map from $U\subset \R^m$ to $\R^n$.
Suppose that the differential $Df:\; T_x\R^m \arrow T_y \R^n$
is injective for one choice of the charts $U,V$.
Prove that it is injective for any other choice
of the charts.
\ez

\definition A smooth map of manifolds
$M\stackrel f \arrow N$ is called {\bf immersion} if its
differential $Df:\; T_xM \arrow T_{f(x)}N$, computed in local
coordinates,  is injective.
\ed



\exercise Construct an immersion of the Klein bottle into
$\Bbb R^3$.
\ez

\exercise[!] Let $M\stackrel f\longrightarrow N$ be a
smooth map of manifolds, where $M$ is compact. 
Show that $f$ is a smooth embedding if and
only if it is an injective immersion.
\ez

\hint Use the inverse function theorem.
\eh

\definition Let $M\hookrightarrow\Bbb R^n$ be a smooth
$m$-submanifold. The {\bf tangent plane} at $p\in M$ is the plane in
$\Bbb R^n$ tangent to $M$ (i.e, the plane lying in the image of the
differential given in local coordinates). A {\bf tangent vector} is an
arbitrary vector in this plane with the origin at $p$. The space of all
tangent vectors at $p$ is denoted by $T_pM$. Given a metric on
$\Bbb R^n$, we can define the space of {\bf unit tangent
vectors} $\Bbb S^{m-1}M$ as the set of all pairs $(p,v)$, where
$p\in M$, $v\in T_pM$, and $|v|=1$.
\ed

\exercise Prove that $\Bbb S^{m-1}M$ is a smooth manifold and that
the natural projection $\Bbb S^{m-1}M\to M$ is a smooth map with fibers
$\Bbb S^{m-1}$.
\ez

\remark $\Bbb S^{m-1}M$ is called the {\bf unit sphere bundle}
over $M$.
\er

\exercise[*] Show that 
the manifold
$\Bbb S^{m-1}M$ does not depend
on an embedding $M\hookrightarrow\Bbb R^n$, i.e., for any two 
embeddings of $M$ into $\Bbb R^n$ and into $\Bbb R^{n'}$, the
corresponding manifolds $\Bbb S^{m-1}M$ are diffeomorphic.
\ez

\exercise[!] Let
$M\stackrel\varphi\hookrightarrow\Bbb  R^n$ be a manifold of
dimension $m$ embedded into $\Bbb R^n$, 
$\lambda\in\Bbb P_\Bbb R^{n-1}$ a straight line in $\Bbb R^n$, and
let $P_\lambda:\Bbb R^n\to\Bbb R^{n-1}$ denote the projection onto the
quotient $\Bbb R^n/\lambda\cong\Bbb R^{n-1}$.
\enum
\ite Denote the diagonal by $\Delta\subset M\times M$. Define the map
$M\times M\setminus\Delta\stackrel B\longrightarrow\Bbb P_\Bbb
R^{n-1}$
by sending the pair of points $(x,y)\in M\times M$ to the straight line
passing through $\varphi(x)-\varphi(y)$. Show that
$P_\lambda\circ\varphi:M\to\Bbb R^{n-1}$ is an injection if and only
if $\lambda$ does not lie in the image of $B$.

\ite
 Let $\Bbb S^{m-1}M\stackrel {B_0}\arrow\Bbb P_\Bbb R^{n-1}$ 
be a map
sending a tangent vector to the 
corresponding line in $\R^n$. Show that
$P_\lambda\circ\varphi:M\to\Bbb R^{n-1}$ is an immersion if and only if
$\lambda$ does not lie in the image of $B_0$.
\ee
\ez

\exercise[!] Let
$M\stackrel\varphi\hookrightarrow\Bbb  R^n$ be an embedded manifold
of dimension $m$ with $n>2m+2$. Show that there exists a projection
$\Bbb R^n\stackrel P\longrightarrow\Bbb R^{2m+2}$ such that
$P\circ\varphi:M\to\Bbb R^{2m+2}$ is an immersion.
\ez

\hint Use the fact that the images of the maps $B_0$ and $B$ in
the previous problem have measure zero and apply induction on $n$.
\eh

\exercise In assumptions of the previous exercise, prove
that there exists a projection
$\Bbb R^n\stackrel P\longrightarrow\Bbb R^{2m+1}$ such that
$P\circ\varphi:M\to\Bbb R^{2m+1}$ is an immersion.
\ez

\exercise Is any $n$-dimensional manifold embeddable in
$\Bbb R^{2n-1}$ ?
\ez

\exercise[**] Is it possible to construct an immersion of
the complex projective space $\Bbb P_\Bbb C^2$ into $\Bbb R^5$ ?
\ez

\exercise
Let $M$ be a compact manifold of
dimension $n$. Show that $M$ admits a smooth closed embedding into
$\Bbb R^{2n+2}$.
\ez

\remark Whitney showed that any $m$-dimensional
manifold with a countable basis of topology admits a closed embedding
into $\Bbb R^{2m}$. This statement is called the ``strong Whitney
theorem.''
\er


\end{document}