Contents Preface ix Chapter 1. Introduction to Fractals 1 Chapter 2. Dimension 11 Chapter 3. Trees and Fractals 15 Chapter 4. Invariant Sets 21 Chapter 5. Probability Trees 23 Chapter 6. Galleries 27 Chapter 7. Probability Trees Revisited 31 Chapter 8. Elements of Ergodic Theory 33 Chapter 9. Galleries of Trees 35 Chapter 10. General Remarks on Markov Systems 37 Chapter 11. Markov Operator T and Measure Preserving Transformation T 39 Chapter 12. Probability Trees and Galleries 43 Chapter 13. Ergodic Theorem and the Proof of the Main Theorem 47 Chapter 14. An Application: The k-lane property 51 Chapter 15. Dimension and Energy 53 Chapter 16. Dimension Conservation 55 Chapter 17. Ergodic Theorem for Sequences of Functions 57 Chapter 18. Dimension Conservation for Homogeneous Fractals: The Main Steps in the Proof 59 Chapter 19. Verifying the Conditions of the Ergodic Theorem for Sequences of Functions 65 Bibliography 67 Index 69 vii
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