Meromorphic Dynamics 2 Volume Hardback Set

Details key results from ergodic theory and geometric measure theory, then applies those techniques to the dynamics of elliptic functions.

Janina Kotus (Author), Mariusz Urba?ski (Author)

9781009216050, Cambridge University Press

Multiple-component retail product, published 4 May 2023

400 pages
23.5 x 15.7 x 6 cm, 1.72 kg

This two-volume set provides a comprehensive and self-contained approach to the dynamics, ergodic theory, and geometry of elliptic functions mapping the complex plane onto the Riemann sphere. Volume I discusses many fundamental results from ergodic theory and geometric measure theory in detail, including finite and infinite abstract ergodic theory, Young's towers, measure-theoretic Kolmogorov-Sinai entropy, thermodynamics formalism, geometric function theory, various conformal measures, conformal graph directed Markov systems and iterated functions systems, classical theory of elliptic functions. In Volume II, all these techniques, along with an introduction to topological dynamics of transcendental meromorphic functions, are applied to describe the beautiful and rich dynamics and fractal geometry of elliptic functions. Much of this material is appearing for the first time in book or even paper form. Both researchers and graduate students will appreciate the detailed explanations of essential concepts and full proofs provided in what is sure to be an indispensable reference.

Volume I. Preface
Acknowledgments
Introduction
Part I. Ergodic Theory and Geometric Measures: 1. Geometric measure theory
2. Invariant measures: finite and infinite
3. Probability (finite) invariant measures: basic properties and existence
4. Probability (finite) invariant measures: finer properties
5. Infinite invariant measures: finer properties
6. measure- theoretic entropy
7. Thermodynamic formalism
Part II. Complex Analysis, Conformal Measures, and Graph Directed Markov Systems: 8. Selected topics from complex analysis
9. Invariant measures for holomorphic maps f in A(X) or in Aw(X)
10. Sullivan conformal measures for holomorphic maps f in A(X) and in Aw(X)
11. Graph directed Markov systems
12. Nice sets for analytic maps
References
Index of symbols
Subject index
Volume II. Preface
Acknowledgments
Introduction
Part III. Topological Dynamics of Meromorphic Functions: 13. Fundamental properties of meromorphic dynamical systems
14. Finer properties of fatou components
15. Rationally indifferent periodic points
Part IV. Elliptic Functions: Classics, Geometry, and Dynamics: 16. Classics of elliptic functions: selected properties
17. Geometry and dynamics of (all) elliptic functions
Part V. Compactly Nonrecurrent Elliptic Functions: First Outlook: 18. Dynamics of compactly norecurrent elliptic functions
19. Various examples of compactly nonrecurrent elliptic functions
Part VI. Compactly Nonrecurrent Elliptic Functions: Fractal Geometry, Stochastic Properties, and Rigidity: 20. Sullivan h-conformal measures for compactly nonrecurrent elliptic functions
21. Hausdorff and packing measures of compactly nonrecurrent regular elliptic functions
22. Conformal invariant measures for compactly nonrecurrent regular elliptic functions
23. Dynamical rigidity of compactly nonrecurrent regular elliptic functions
Appendix A. A quick review of some selected facts from complex analysis of a one-complex variable
Appendix B. Proof of the Sullivan nonwandering theorem for speiser class S
References
Index of symbols
Subject index.

Subject Areas: Stochastics [PBWL], Fractal geometry [PBMX], Geometry [PBM], Complex analysis, complex variables [PBKD], Mathematics [PB]