Preface

Statistical Mechanics and Quantum Fields on Fractals

1. Introduction

2. Discrete scaling symmetry - Self similarity - Definitions

3. Heat kernel and spectral functions - Generalities

4. Laplacian on fractals - Heat kernel and spectral zeta function

5. Thermodynamics on photons : The fractal blackbody [34]

6. Conclusion and some open questions

Acknowledgments

References

Spectral Algebra of the Chernov and Bogoslovsky Finsler Metric Tensors

Preliminaries

1. Spectral theory prerequisites

2. Spectral results for low dimensions

3. Conclusions

References

Local Multifractal Analysis

1. Introduction

2. Properties of the local Hausdorff dimension and the local multifractal spectrum

3. A local multifractal formalism for a dyadic family

4. Measures with varying local spectrum

5. Local spectrum of stochastic processes

6. Other regularity exponents characterized by dyadic families

7. A functional analysis point of view

Acknowledgement

References

Extreme Risk and Fractal Regularity in Finance

1. Introduction

2. Fractal Regularities in Financial Markets

3. The Markov-Switching Multifractal (MSM)

4. Pricing Multifractal Risk

5. Conclusion

References

An Algorithm for Dynamical Games with Fractal-Like Trajectories

1. Introduction

2. Preliminaries and notations

3. The method for 𝐶¹ games

4. Two players parametric games

5. The algorithm

6. Examples

7. Final Remarks

8. Resume

9. Conclusions

References

The Landscape of Anderson Localization in a Disordered Medium

1. Introduction

2. Preliminaries

3. The control inequalities

4. Numerical simulations

5. Conclusion

References

Zeta Functions for Infinite Graphs and Functional Equations

0. Introduction

1. Zeta functions for infinite graphs

2. Functional equations for infinite graphs

References

Vector Analysis on Fractals and Applications

1. Introduction

2. Dirichlet forms and energy measures

3. 1-forms and vector fields

4. Scalar PDE involving first order terms

5. Navier-Stokes equations

6. Magnetic Schrödinger equations

References

Non-Regularly Varying and Non-Periodic Oscillation of the On-Diagonal Heat Kernels on Self-Similar Fractals

1. Introduction

2. Framework and main results

3. Proof of Theorems 2.17 and 2.18

4. Post-critically finite self-similar fractals

4.1. Harmonic structures and resulting self-similar Dirichlet spaces

4.2. Cases with good symmetry and affine nested fractals

4.3. Cases possibly without good symmetry

5. Sierpiński carpets

References

Lattice Effects in the Scaling Limit of the Two-Dimensional Self-Avoiding Walk

1. Introduction

2. Lattice effects

3. Simulations

4. Conclusions and future work

References

The Casimir Effect on Laakso Spaces

1. Introduction

2. Laakso spaces

3. Spectral Zeta Functions

4. Casimir Effect

5. Finite Approximations to Laakso Spaces

6. Casimir Effect on L

7. A Higher Dimensional Case

References

The Decimation Method for Laplacians on Fractals: Spectra and Complex Dynamics

1. Introduction

2. The bounded Sierpinski gasket

2.1. Spectral properties of the Laplacian on the Sierpinski gasket

3. Generalization of the decimation method

3.1. The fractal Sturm–Liouville operator

3.2. The eigenvalue problem

3.3. The renormalization map and the spectrum of the operator

4. An infinite lattice based on the Sierpinski gasket

5. Factorization of the spectral zeta function

References

The Current State of Fractal Billiards

1. Introduction

2. Rational billiards

2.1. Translation surfaces and properties of the flow

2.2. Unfolding a billiard orbit and equivalence of flows

3. The fractals of interest

3.1. Cantor sets

3.2. The Koch curve and Koch snowflake

3.3. The 𝑇-fractal

3.4. Self-similar Sierpinski carpets

4. Prefractal (rational) billiards

4.1. A general structure

4.2. The prefractal Koch snowflake billiard

4.3. The 𝑇-fractal prefractal billiard

4.4. A prefractal self-similar Sierpinski carpet billiard

5. Fractal billiards

5.1. A general framework for Ω(𝐾𝑆), Ω(𝒯) and Ω(𝒮ₐ)

5.2. The Koch snowflake fractal billiard

5.3. The 𝑇-fractal billiard

5.4. A self-similar Sierpinski carpet billiard

6. Concluding remarks

References

Long-Range Dependence and the Rank of Decompositions

1. Introduction

2. The Gaussian case

3. The linear case: Surgailis approach

4. The linear case: Ho and Hsing approach

5. Application to the polynomial case

6. Sketches of proofs of Theorems 2.2 , 3.2 and 4.2

7. Conclusion

Acknowledgments

References

Hitting Probabilities of the Random Covering Sets

Introduction

1. Main results and examples

2. Proofs of the theorems

3. Technical results

References

Fractal Oscillations Near the Domain Boundary of Radially Symmetric Solutions of 𝑝-Laplace Equations

1. Introduction

2. A bi-Lipschitz transformation of equation (1.12)

3. Qualitative properties of equations (1.12) and (2.10)

4. Proof of Theorem 1.2

5. Proof of Proposition 1.1

6. Proof of Lemma 1.1

Acknowledgements

References

Applications of the Contraction Mapping Principle

1. The Contraction Mapping Principle

2. Corollaries, Applications and Implications

3. Fractal Method of Solutions to Inverse Problems of ODEs

4. Self-Similarity

5. A Derivative Corresponding to the Box-Counting Dimension

6. Representation Theory of Fractal Sets

7. Spacelike Cantor Sets in a Toy Model

8. Concluding Remarks and Future Directions

References

Economics and Psychology. Perfect Rationality versus Bounded Rationality

Introduction

1. Economics and the ‘perfect’ rationality

2. Psychology into Economics. The cognitive dimension.

3. Behavioral Finance

4. Bounded rationality and risk aversion: a model of behavioral finance

Conclusions

References