Page 263 - DMTH503_TOPOLOGY

Unit 32: Introduction to Dimension Theory

Consider a probability vector p = (p , …, p ) and define the Bernoulli measure of any level Notes
0 n–1
cylinder to be,

 [ ( i , ,i k 1 ]) = p k 0 0 ( i ) p k 1 1 ( i ) p k - n 1 ( i ) .
n 1
0
-
-
A probability measure  on  is said to be invariant under the shift map if for any Borel set
–1
–1
B  X, (B) =( (B)). We say thatis ergodic if any Borel set Bsuch that  (X) = X satisfies
(X) = 0 or (X) = 1. A Bernoulli measure is both invariant and ergodic.
32.2 Summary
 Criteria for defining a dimension
(i) When X is a manifold then the value of the dimension is an integer which coincides
with the usual notion of dimension;

(ii) For more general sets X we can have “fractional” dimensional; and
(iii) Points and countable unions of points, have zero dimension.
 For a given probability measure , we define the Hausdorff dimension of the measure by

dim () = inf {dim (X) : (X) = 1}.
H H
 The upper and lower pointwise dimensions of a measure  are measurable functions, d ,

d : X{¥} defined by

log (B(x,r))

d (x) = lim sup and

r 0 log r
log (B(x,r))

d (x) = lim inf

r 0 log r
32.3 Keywords

Countable Set: A set is countable if it is non-empty and finite or if it is countably infinite.
Hausdorff Space: A topological space (X, T) is called Hausdorff space if given a pair of distinct
points x, yX,
G, HT s.t. xG, yH, GÇH = .
Iterated Function Scheme: An iterated function scheme on an open setR consists of a family
d
of contractions T , …, T :  .
1 k
Open Set: Any set AT is called an open set.
Subcover: Let (X, T) be a topological space and AX. Let G denote a family of subsets of X. If
G G s.t. G is a finite set and that {G : GG } is a cover of A then G is called a finite subcover
1 1 1 1
of the original cover.
32.4 Review Questions

1. Write a short note on Dimension Theory.
2. State Besicovitch covering lemma.

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