Expanding Thurston Maps
Seiten
2018
American Mathematical Society (Verlag)
978-0-8218-7554-4 (ISBN)
American Mathematical Society (Verlag)
978-0-8218-7554-4 (ISBN)
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A monograph devoted to the study of the dynamics of expanding Thurston maps under iteration. A Thurston map is a branched covering map on a two-dimensional topological sphere such that each critical point of the map has a finite orbit under iteration.
This monograph is devoted to the study of the dynamics of expanding Thurston maps under iteration. A Thurston map is a branched covering map on a two-dimensional topological sphere such that each critical point of the map has a finite orbit under iteration. A Thurston map is called expanding if, roughly speaking, preimages of a fine open cover of the underlying sphere under iterates of the map become finer and finer as the order of the iterate increases. Every expanding Thurston map gives rise to a fractal space, called its visual sphere. Many dynamical properties of the map are encoded in the geometry of this visual sphere. For example, an expanding Thurston map is topologically conjugate to a rational map if and only if its visual sphere is quasisymmetrically equivalent to the Riemann sphere. This relation between dynamics and fractal geometry is the main focus for the investigations in this work.
This monograph is devoted to the study of the dynamics of expanding Thurston maps under iteration. A Thurston map is a branched covering map on a two-dimensional topological sphere such that each critical point of the map has a finite orbit under iteration. A Thurston map is called expanding if, roughly speaking, preimages of a fine open cover of the underlying sphere under iterates of the map become finer and finer as the order of the iterate increases. Every expanding Thurston map gives rise to a fractal space, called its visual sphere. Many dynamical properties of the map are encoded in the geometry of this visual sphere. For example, an expanding Thurston map is topologically conjugate to a rational map if and only if its visual sphere is quasisymmetrically equivalent to the Riemann sphere. This relation between dynamics and fractal geometry is the main focus for the investigations in this work.
Mario Bonk, University of California, Los Angeles, CA. Daniel Meyer, University of Liverpool, UK.
Introduction
Thurston maps
Lattes maps
Quasiconformal and rough geometry
Cell decompositions
Expansion
Thurston maps with two or three postcritical points
Visual metrics
Symbolic dynamics
Tile graphs
Isotopies
Subdivisions
Quotients of Thurston maps
Combinatorially expanding Thurston maps
Invariant curves
The combinatorial expansion factor
The measure of maximal entropy
The geometry of the visual sphere
Rational Thurston maps and Lebesgue measure
A combinatorial characterization of Lattes maps
Outlook and open problems
Appendix A
Bibliography
Index.
| Erscheinungsdatum | 08.12.2017 |
|---|---|
| Reihe/Serie | Mathematical Surveys and Monographs |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 1000 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Geometrie / Topologie |
| ISBN-10 | 0-8218-7554-X / 082187554X |
| ISBN-13 | 978-0-8218-7554-4 / 9780821875544 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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