*Araceli Bonifant, Mikhail Lyubich, Scott Sutherland, Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.003.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This introductory chapter is an overview into holomorphic dynamics—one of the earliest branches of dynamical systems which is not part of classical mechanics. Holomorphic dynamics studies iterates of ...
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This introductory chapter is an overview into holomorphic dynamics—one of the earliest branches of dynamical systems which is not part of classical mechanics. Holomorphic dynamics studies iterates of holomorphic maps on complex manifolds. As a prominent field in its own right, holomorphic dynamics was founded early in the twentieth century, but was almost completely forgotten for sixty years, only to be revived in the early 1980s partly due to the efforts of John Milnor. The field of holomorphic dynamics is rich in interactions with many branches of mathematics; such as complex analysis, geometry, topology, number theory, algebraic geometry, combinatorics, and measure theory. This chapter briefly explores the extent of such interplay.Less

This introductory chapter is an overview into holomorphic dynamics—one of the earliest branches of dynamical systems which is not part of classical mechanics. Holomorphic dynamics studies iterates of holomorphic maps on complex manifolds. As a prominent field in its own right, holomorphic dynamics was founded early in the twentieth century, but was almost completely forgotten for sixty years, only to be revived in the early 1980s partly due to the efforts of John Milnor. The field of holomorphic dynamics is rich in interactions with many branches of mathematics; such as complex analysis, geometry, topology, number theory, algebraic geometry, combinatorics, and measure theory. This chapter briefly explores the extent of such interplay.

*Araceli Bonifant, Misha Lyubich, and Scott Sutherland*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.001.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, ...
More

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing.Less

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing.

*Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.003.0024
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter surveys developments arising from John Milnor's 1958 paper, “On the existence of a connection with curvature zero” and his 1977 paper, “On fundamental groups of complete affinely flat ...
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This chapter surveys developments arising from John Milnor's 1958 paper, “On the existence of a connection with curvature zero” and his 1977 paper, “On fundamental groups of complete affinely flat manifolds.” The former deeply influenced the theory of characteristic classes of flat bundles, and the latter clarified the theory of affine manifolds, setting the stage for its future flourishing. This chapter begins with some reminiscences on Milnor. It then describes the history of the Milnor–Wood inequality and the Auslander Conjecture and then proceeds to more recent developments, including a description of Margulis space-times, a startling example of an affine three-manifold with free fundamental group.Less

This chapter surveys developments arising from John Milnor's 1958 paper, “On the existence of a connection with curvature zero” and his 1977 paper, “On fundamental groups of complete affinely flat manifolds.” The former deeply influenced the theory of characteristic classes of flat bundles, and the latter clarified the theory of affine manifolds, setting the stage for its future flourishing. This chapter begins with some reminiscences on Milnor. It then describes the history of the Milnor–Wood inequality and the Auslander Conjecture and then proceeds to more recent developments, including a description of Margulis space-times, a startling example of an affine three-manifold with free fundamental group.