Dimensions, Embeddings, and Attractors

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
Introduction......Page 15
PART I: Finite-dimensional sets......Page 19
1 Lebesgue covering dimension......Page 21
1.1 Covering dimension......Page 22
1.2 The covering dimension of In......Page 24
1.3 Embedding sets with finite covering dimension......Page 26
1.4 Large and small inductive dimensions......Page 31
Exercises......Page 32
2.1 Hausdorff measure and Lebesgue measure......Page 34
2.2 Hausdorff dimension......Page 37
2.3 The Hausdorff dimension of products......Page 39
2.4 Hausdorff dimension and covering dimension......Page 40
Exercises......Page 43
3.1 The definition of the box-counting dimension......Page 45
3.2 Basic properties of the box-counting dimension......Page 47
3.3 Box-counting dimension of products......Page 49
3.4 Orthogonal sequences......Page 50
Exercises......Page 53
4 An embedding theorem for subsets of RN in terms of the upper box-counting dimension......Page 55
5.1 Prevalence......Page 61
5.2 Measures based on sequences of linear subspaces......Page 63
5.2.1 The probe set and its measure in a Hilbert space......Page 64
5.2.2 The probe set and its measure in a Banach space......Page 67
Exercises......Page 70
6 Embedding sets with dH(X - X) finite......Page 71
6.1 No linear embedding is possible when dH(X) is finite......Page 72
6.2 Embedding sets with dH(X−X) finite......Page 74
6.3 No modulus of continuity is possible for L−1......Page 76
7 Thickness exponents......Page 78
7.1 The thickness exponent......Page 79
7.2 Lipschitz deviation......Page 81
7.2.1 An example with dev…......Page 82
7.3 Dual thickness......Page 83
Exercises......Page 87
8.1 Embedding sets with Holder continuous parametrisation......Page 89
8.2 Sharpness of the Holder exponent......Page 91
Exercises......Page 95
9.1 Homogeneous spaces and the Assouad dimension......Page 97
9.2 Assouad dimension and products......Page 100
9.3 Orthogonal sequences......Page 102
9.4 Homogeneity is not sufficient for a bi-Lipschitz embedding......Page 105
9.5 Almost bi-Lipschitz embeddings......Page 108
9.6 Sharpness of the logarithmic exponent......Page 113
Exercises......Page 114
PART II: Finite-dimensional attractors......Page 117
10.1 Nonlinear semigroups and attractors......Page 119
10.2 Sobolev spaces and fractional power spaces......Page 120
10.3 Abstract semilinear parabolic equations......Page 122
10.4 The two-dimensional Navier–Stokes equations......Page 123
Exercises......Page 127
11.2 Existence of the global attractor......Page 129
11.3 Example 1: semilinear parabolic equations......Page 132
11.4 Example 2: the two-dimensional Navier–Stokes equations......Page 133
Exercises......Page 135
12 Bounding the box-counting dimension of attractors......Page 137
12.1 Coverings of T[B(0, 1)] via finite-dimensional approximations......Page 139
12.2 A dimension bound when…......Page 143
12.4 Semilinear parabolic equations in Hilbert spaces......Page 144
Exercises......Page 146
13.1 Zero thickness......Page 150
13.2 Zero Lipschitz deviation......Page 152
Exercises......Page 157
14.1 The finite-dimensional case......Page 159
14.2 Periodic orbits and the Lipschitz constant for ordinary differential equations......Page 166
14.3 The infinite-dimensional case......Page 168
14.4 Periodic orbits and the Lipschitz constant for semilinear parabolic equations......Page 170
Exercises......Page 172
15 Parametrisation of attractors via point values......Page 174
15.1.1 Real analytic functions......Page 175
15.1.2 Order of vanishing......Page 176
15.2 Dimension and thickness of…......Page 177
15.3 Proof of Theorem 15.1......Page 179
15.4.1 Determining nodes......Page 181
15.4.2 Degrees of freedom in turbulent flows......Page 182
Exercises......Page 183
Solutions to exercises......Page 184
References......Page 210
Index......Page 216