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## Hyperbolic Dynamics and Brownian Motion: An Introduction

*Jacques Franchi and Yves Le Jan*

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199654109
- eISBN:
- 9780191745676
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199654109.001.0001
- Subject:
- Mathematics, Mathematical Physics

The idea of this book is to illustrate an interplay between distinct domains of mathematics. Firstly, this book provides an introduction to hyperbolic geometry, based on the Lorentz group PSO(1, d) ... More

## A framework toward understanding the characterization of holomorphic dynamics

*Yunping Jiang*

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

### in Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday

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

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.003.0012
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter reviews the characterization of geometrically finite rational maps and then outlines a framework for characterizing holomorphic maps. Whereas Thurston's methods are based on estimates of ... More

## Curved Space

*Nicholas Manton and Nicholas Mee*

### in The Physical World: An Inspirational Tour of Fundamental Physics

- Published in print:
- 2017
- Published Online:
- July 2017
- ISBN:
- 9780198795933
- eISBN:
- 9780191837111
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198795933.003.0006
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter develops the mathematical technology required to understand general relativity by taking the reader from the traditional flat space geometry of Euclid to the geometry of Riemann that ... More

## Curves, Surfaces, and Hyperbolic Geometry

*Benson Farb and Dan Margalit*

### in A Primer on Mapping Class Groups (PMS-49)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691147949
- eISBN:
- 9781400839049
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691147949.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter explains the basics of working with simple closed curves, focusing on the case of the closed surface Sɡ of genus g. When g is greater than or equal to 2, hyperbolic geometry enters as a ... More

## Mappings and moduli

*Simon Donaldson*

### in Riemann Surfaces

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198526391
- eISBN:
- 9780191774874
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526391.003.0014
- Subject:
- Mathematics, Geometry / Topology, Analysis

This chapter analyzes diffeomorphisms of the plane; braids, Dehn twists and quadratic singularities; hyperbolic geometry; and compactification of the moduli space.

## Hyperbolic geometry

*Jacques Franchi and Yves Le Jan*

### in Hyperbolic Dynamics and Brownian Motion: An Introduction

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199654109
- eISBN:
- 9780191745676
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199654109.003.0002
- Subject:
- Mathematics, Mathematical Physics

Basic notions of hyperbolic geometry are presented: geodesics, light rays, tangent bundles etc. These are systematically derived from properties of Minkowski space: the boundary of hyperbolic space ... More

## The Ping-Pong Lemma

*Johanna Mangahas*

### in Office Hours with a Geometric Group Theorist

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691158662
- eISBN:
- 9781400885398
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691158662.003.0005
- Subject:
- Mathematics, Geometry / Topology

This chapter considers an identifying feature of free groups: their ability to play ping-pong. In mathematics, you may encounter a group without immediately knowing which group it is. Fortunately, ... More

## The Nielsen-Thurston Classification

*Benson Farb and Dan Margalit*

### in A Primer on Mapping Class Groups (PMS-49)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691147949
- eISBN:
- 9781400839049
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691147949.003.0014
- Subject:
- Mathematics, Geometry / Topology

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of ... More

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