Prologue ........................................................ 1
0.1. Introduction ............................................ 1
0.2. the Unit square ......................................... 4
Chapter 1. Scales and Volume Doubling Property of Measures ...... 9
1.1. Scale ................................................... 9
1.2. Self-similar structures and measures ................... 13
1.3. Volume doubling property ............................... 16
1.4. Locally finiteness and gentleness ...................... 20
1.5. Rationally ramified self-similar sets 1 ................ 23
1.6. Rationally ramified self-similar sets 2 ................ 29
1.7. Examples ............................................... 34
Chapter 2. Construction of Distances ........................... 43
2.1. Distances associated with scales ....................... 43
2.2. Intersection type ...................................... 46
2.3. Qdistances adapted to scales ........................... 51
Chapter 3. Heat Kernel and Volume Doubling Property of
Measures ............................................ 59
3.1. Dirichlet forms on self-similar sets ................... 59
3.2. Heat kernel estimate ................................... 63
3.3. P. с. f. self-similar sets ............................. 64
3.4. Sierpinski carpets ..................................... 70
3.5. Proof of Theorem 3.2.3 ................................. 74
Appendix ....................................................... 83
A. Existence and continuity of a heat kernel ................ 83
B. Recurrent case and resistance form ....................... 86
C. Heat kernel estimate to the volume doubling property ..... 87
Bibliography ................................................... 89
Assumptions, Conditions and Properties in Parentheses .......... 91
List of Notations .............................................. 92
Index .......................................................... 93
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