Page 259 - DMTH503_TOPOLOGY

Richa Nandra, Lovely Professional University Unit 32: Introduction to Dimension Theory

Unit 32: Introduction to Dimension Theory Notes

CONTENTS
Objectives
Introduction

32.1 Introduction to Dimension Theory
32.1.1 Hausdorff Dimension of Measures
32.1.2 Pointwise Dimension

32.1.3 Besicovitch Covering Lemma
32.1.4 Bernoulli’s Measures
32.2 Summary
32.3 Keywords
32.4 Review Questions

32.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Know about the dimensional theory;
 Define Hausdorff dimension of measures;

 Define pointwise dimension;
 Solve the problems on the dimensional theory.

Introduction

For many familiar objects there is a perfectly reasonable intuitive definition of dimension: A
d
space is d-dimensional if locally it looks like a patch R . This immediately allows us to say: The
dimension of a point is zero; the dimension of a line is 1; the dimension of a plane is 2; the
dimension of R is d.
d
There are several different notions of dimension for more general sets, some more easy to
compute and others more convenient in applications. We shall concentrate on Hausdorff
dimension. Hausdorff introduced his definition of dimension in 1919. Further contributions and
applications, particularly to number theory, were made by Besicovitch.

Hausdorff’s idea was to find the value at which the measurement changes from infinite to zero.
Dimension is at the heart of all fractal geometry, and provides a reasonable basis for an invariant
between different fractal objects.
32.1 Introduction to Dimension Theory

Before we begin defining Hausdorff and other dimensions, it is a good idea to clearly state our
objectives. What should be the features of a good definition of dimension? Based on intuition,

LOVELY PROFESSIONAL UNIVERSITY 253