Page 260 - DMTH503_TOPOLOGY

Topology

Notes we would expect that the dimension of an object would be related to its measurement at a certain
scale. For example, when an object is scaled by a factor of 2.
1
 for a line segment, its measure will increase by 2 = 2
2
 for a rectangle, its measures will increase by 2 = 4
3
 for a parallelepiped, its measures will increase by 2 = 8
In each case, we extract the exponent and consider this to be the dimension. More precisely, dim
F = log (F)/log 1/p where p is the precision (1/p is the scaling factor) and (F) is the change
in the ‘measure’ of F when scaled by 1/p. Falconer suggests that most of following criteria also
2
be net [Falc ], by any thing called a dimension:
1. Smooth manifolds: If F is any smooth, n-dimensional manifold, dim F = n.
n
2. Open Sets: For an open subset F   , dim F = n.
3. Countable Sets: dim F = 0 if F is finite or countable.

4. Monotonicity: E  Fdim Edim F.
5. Stability: dim (EF) = max (dim E, dim F).

6. Countable Stability: dim( ¥ i 1 F i ) = sup {dimF }.
=
i
i
m
7. Lipschitz Mapping: If f : E is lipschitz, then dim f(E)dim (E).
8. Bi-lipschitz Mapping: If f : E is Bi-lipschitz, then dim f(E) = dim (E).
m
9. Geometric Invariance: dim f(F) = dim F, if f is a similarity or affine transformation.
Recall that f : E is Lipschitz iffc such that
m
|f(x) – f(y)| c|x – y|  x, y  E;

and that f is Bi-lipschitz iffc , c such that
1 2
c |x – y| |f(x) – f(y)| c |x – y|  x, y  E;
1 2
and f is a Similarity iffc such that

|f(x) – f(y)| = c|x – y|  x, y  E;
32.1.1 Hausdorff Dimension of Measures

Let  denote a probability measure on a set of X. We can define the Hausdorff dimension  in
terms of the Hausdorff dimension of subsets of A.
Definition: For a given probability measure  we define the Hausdorff dimension of the measure
by
dim () = inf {dim (X) :  (X) = 1}.
H H
We next want to define a local notion of dimension for a measure  at a typical point xX.

32.1.2 Pointwise Dimension

Definition: The upper and lower pointwise dimensions of a measure  are measurable functions,


log (B(x,r)) log (B(x,r))
d , d : X  R  { } defined by d (x) = lim sup and d (x) = lim inf
¥
   
r 0 log r r 0 log r
where B(x, r) is a ball of radius r > 0 about x.
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