Janina Kotus is Professor of Mathematics at the Warsaw University of Technology, Poland. Her research focuses on dynamical systems, in particular holomorphic dynamical systems. Together with I. N. Baker and Y. Lű she laid the foundations for iteration of meromophic functions. Mariusz Urbański is Professor of Mathematics at the University of North Texas. His research interests include dynamical systems, ergodic theory, fractal geometry, iteration of rational and meromorphic functions, open dynamical systems, iterated function systems, Kleinian groups, diophantine approximations, topology and thermodynamic formalism. He is the author of eight books, seven monographs, and more than 200 papers.

• 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.

Janina Kotus is Professor of Mathematics at the Warsaw University of Technology, Poland. Her research focuses on dynamical systems, in particular holomorphic dynamical systems. Together with I. N. Baker and Y. Lű she laid the foundations for iteration of meromophic functions. Mariusz Urbański is Professor of Mathematics at the University of North Texas. His research interests include dynamical systems, ergodic theory, fractal geometry, iteration of rational and meromorphic functions, open dynamical systems, iterated function systems, Kleinian groups, diophantine approximations, topology and thermodynamic formalism. He is the author of eight books, seven monographs, and more than 200 papers.

• 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.

Janina Kotus is Professor of Mathematics at the Warsaw University of Technology, Poland. Her research focuses on dynamical systems, in particular holomorphic dynamical systems. Together with I. N. Baker and Y. Lű she laid the foundations for iteration of meromophic functions. Mariusz Urbański is Professor of Mathematics at the University of North Texas. His research interests include dynamical systems, ergodic theory, fractal geometry, iteration of rational and meromorphic functions, open dynamical systems, iterated function systems, Kleinian groups, diophantine approximations, topology and thermodynamic formalism. He is the author of eight books, seven monographs, and more than 200 papers.

• 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.