Local Dynamics of Non-Invertible Maps near Normal Surface Singularities.

Cover
Title page
Introduction
Chapter 1. Normal surface singularities, resolutions, and intersection theory
1.1. The intersection theory of good resolutions
1.2. Log resolutions and divisors
Chapter 2. Normal surface singularities and their valuation spaces
2.1. Classification of finite semivaluations
2.2. Dual divisors associated to valuations and b-divisors
2.3. Intersection theory and skewness
2.4. Weak topology and tangent vectors
2.5. Dual graphs and the structure of \mc{ }_{ }
2.6. Partial order, trees, and parameterizations
2.7. The angular metric
Chapter 3. Log discrepancy, essential skeleta, and special singularities
3.1. Log canonical and log terminal singularities
3.2. The essential skeleton
Chapter 4. Dynamics on valuation spaces
4.1. Induced maps on valuation spaces
4.2. Action on dual divisors
4.3. Action on b-divisors
4.4. Angular distance is non-increasing
4.5. The Jacobian formula
4.6. Critical skeleton
4.7. Classification of valuative dynamics
Chapter 5. Dynamics of non-finite germs
5.1. Construction of an eigenvaluation
5.2. Weak convergence
5.3. Semi-superattracting germs
5.4. Strong convergence
Chapter 6. Dynamics of non-invertible finite germs
6.1. Quotient singularities
6.2. Non-lt singularities
6.3. Irrational rotations on cusp singularities
Chapter 7. Algebraic stability
7.1. Existence of geometrically stable models
7.2. Smoothness of geometrically stable models
Chapter 8. Attraction rates
8.1. First dynamical degree
8.2. Recursion relations for the sequence of attraction rates
8.3. Finite germs on cusp singularities
Chapter 9. Examples and remarks
9.1. A finite map at a smooth point
9.2. A non-finite map at a smooth point
9.3. A quotient singularity
9.4. A simple elliptic singularity.

9.5. Quasihomogeneous singularities
9.6. A non-finite map on a cusp singularity
9.7. A finite map on a cusp singularity
9.8. Different normalizations
9.9. Automorphisms
9.10. Positive characteristic
Appendix A. Cusp singularities
A.1. Arithmetic construction of cusp singularities
A.2. Finite endomorphisms
Bibliography
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