This book contains a rigorous mathematical treatment of the geometrical aspects of sets of both integral and fractional Hausdorff dimension. Questions of local density and the existence of tangents of such sets are studied, as well as the dimensional properties of their projections in various directions. In the case of sets of integral dimension the dramatic differences between regular 'curve-like' sets and irregular 'dust like' sets are exhibited. The theory is related by duality to Kayeka sets (sets of zero area containing lines in every direction). The final chapter includes diverse examples of sets to which the general theory is applicable: discussions of curves of fractional dimension, self-similar sets, strange attractors, and examples from number theory, convexity and so on. There is an emphasis on the basic tools of the subject such as the Vitali covering lemma, net measures and Fourier transform methods.
Author(s): K. J. Falconer
Series: Cambridge Tracts in Mathematics
Publisher: CUP
Year: 1985
Language: English
Pages: 177
Cover......Page 1
Cambridge Tracts in Mathematics 85......Page 2
The geometry of fractal sets......Page 4
Goto 4 /FitH 555521256941......Page 5
Contents......Page 6
Preface......Page 8
Introduction......Page 10
Notation......Page 14
1.1 Basic measure theory......Page 16
1.2 Hausdorff measure......Page 22
1.3 Covering results......Page 25
1.4 Lebesgue measure......Page 27
1.5 Calculation of Hausdorff dimensions and measures......Page 29
2.1 Introduction......Page 35
2.2 Elementary density bounds......Page 37
3.2 Curves and continua......Page 43
3.3 Density and the characterization of regular 1-sets......Page 55
3.4 Tangency properties......Page 61
3.5 Sets in higher dimensions......Page 65
4.2 s-sets with 0 < s < 1......Page 69
4.3 s-sets with s > 1......Page 72
4.4 Sets in higher dimensions......Page 78
5.1 Construction of net measures......Page 79
5.2 Subsets of finite measure......Page 82
5.3 Cartesian products of sets......Page 85
6.2 Hausdorff measure and capacity......Page 90
6.3 Projection properties of sets of arbitrary dimension......Page 95
6.4 Projection properties of sets of integral dimension......Page 99
6.5 Further variants......Page 107
7.1 Introduction......Page 110
7.2 Construction of Besicovitch and Kakeya sets......Page 111
7.3 The dual approach......Page 116
7.4 Generalizations......Page 121
7.5 Relationship with harmonic analysis......Page 124
8.2 Curves of fractional dimension......Page 128
8.3 Self-similar sets......Page 133
8.4 Osculatory and Apollonian packings......Page 140
8.5 An example from number theory......Page 146
8.6 Some applications to convexity......Page 150
8.7 Attractors in dynamical systems......Page 153
8.8 Brownian motion......Page 160
8.9 Conclusion......Page 164
References......Page 165
Index......Page 176
