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LEADER 04150cam a2200553 i 4500

001 ocn871187013;

003 OCoLC;

005 20141212071937.0;

008 140225t20142014riua b 000 0 eng ;

010 a| 2014006823;

020 a| 9781470415228 (alk. paper);

020 a| 1470415224 (alk. paper);

029 1 a| AU@b| 000052593211;

035 a| (Sirsi) o871187013;

035 a| (OCoLC)871187013;

042 a| pcc;

049 a| EREE;

050 00 a| QA691b| .S45 2014;

082 00 a| 515/.482| 23;

084 a| 37E20a| 37E05a| 37E152| msc;

100 1 a| Schwartz, Richard Evane| author.=| ^A716314;

245 14 a| The octagonal PETs /c| Richard Evan Schwartz.;

246 3 a| Octagonal polytope exchange transformations;

264 1 a| Providence, Rhode Island :b| American Mathematical Society,c| [2014];

300 a| x, 212 pages :b| illustrations (some color) ; 26 cm.;

336 a| text2| rdacontent;

337 a| unmediated2| rdamedia;

338 a| volume2| rdacarrier;

490 1 a| Mathematical surveys and monographs ;v| volume 197;

504 a| Includes bibliographical references (pages 211-212).;

505 0 a| 1. Introduction -- 2. Background -- 3. Multigraph PETs -- 4. The alternating grid system -- 5. Outer billiards on semiregular octagons -- 6. Quarter turn compositions -- 7. Elementary properties -- 8. Orbit stability and combinatorics -- 9. Bilateral symmetry -- 10. Proof of the main theorem -- 11. The renormalization map -- 12. Properties of the tiling -- 13. The filling lemma -- 14. The covering lemma -- 15. Further geometric results -- 16. Properties of the limit set -- 17. Hausdorff convergence -- 18. Recurrence relations -- 19. Hausdorff dimension bounds -- 20. Controlling the limit set -- 21. The arc case - - 22. Further symmetries of the tiling -- 23. The forest case -- 24. The cantor set case -- 25. Dynamics in the arc case -- 26. Computational methods -- 27. The calculations -- 28. The raw data.;

520 3 a| A polytope exchange transformation is a (discontinuous) map from a polytope to itself that is a translation wherever it is defined. The 1-dimensional examples, interval exchange transformations, have been studied fruitfully for many years and have deep connections to other areas of mathematics, such as Teichmüller theory. This book introduces a general method for constructing polytope exchange transformations in higher dimensions and then studies the simplest example of the construction in detail. The simplest case is a 1-parameter family of polygon exchange transformations that turns out to be closely related to outer billiards on semi-regular octagons. The 1-parameter family admits a complete renormalization scheme, and this structure allows for a fairly complete analysis both of the system and of outer billiards on semi-regular octagons. The material in this book was discovered through computer experimentation. On the other hand, the proofs are traditional, except for a few rigorous computer-assisted calculations.;

650 0 a| Polytopes.=| ^A46698;

650 0 a| Geometry.=| ^A18295;

650 7 a| Dynamical systems and ergodic theory -- Low-dimensional dynamical systems -- Universality, renormalization.2| msc;

650 7 a| Dynamical systems and ergodic theory -- Low-dimensional dynamical systems -- Maps of the interval (piecewise continuous, continuous, smooth)2| msc;

650 7 a| Dynamical systems and ergodic theory -- Low-dimensional dynamical systems -- Combinatorial dynamics (types of periodic orbits)2| msc;

650 7 a| Geometry.2| fast0| (OCoLC)fst00940864;

650 7 a| Polytopes.2| fast0| (OCoLC)fst01070819;

830 0 a| Mathematical surveys and monographsv| no. 197.=| ^A638333;

938 a| YBP Library Servicesb| YANKn| 11803621;

938 a| Baker and Taylorb| BTCPn| BK0014983772;

949 a| QA691 .S45 2014h| Joyner48i| 30372015660803o| jwjh;

994 a| C0b| ERE;

596 a| 1;

998 a| 3510233;

999 a| QA691 .S45 2014w| LCc| 1i| 30372016761881d| 3/17/2015e| 11/12/2014l| JGESm| JOYNERr| Ys| Yt| JGESBKu| 11/12/2014x| BOOKz| JSTACKSo| .STAFF. jwjh;

The octagonal PETs / Richard Evan Schwartz.

Author/creator	Schwartz, Richard Evan author.
Format	Book
Publication	Providence, Rhode Island : American Mathematical Society, [2014]
Copyright Date	©2014
Description	x, 212 pages : illustrations (some color) ; 26 cm.
Subjects	Polytopes. Geometry.

Variant title	Octagonal polytope exchange transformations
Series	Mathematical surveys and monographs ; volume 197 Mathematical surveys and monographs no. 197. ^A638333
Contents	1. Introduction -- 2. Background -- 3. Multigraph PETs -- 4. The alternating grid system -- 5. Outer billiards on semiregular octagons -- 6. Quarter turn compositions -- 7. Elementary properties -- 8. Orbit stability and combinatorics -- 9. Bilateral symmetry -- 10. Proof of the main theorem -- 11. The renormalization map -- 12. Properties of the tiling -- 13. The filling lemma -- 14. The covering lemma -- 15. Further geometric results -- 16. Properties of the limit set -- 17. Hausdorff convergence -- 18. Recurrence relations -- 19. Hausdorff dimension bounds -- 20. Controlling the limit set -- 21. The arc case - - 22. Further symmetries of the tiling -- 23. The forest case -- 24. The cantor set case -- 25. Dynamics in the arc case -- 26. Computational methods -- 27. The calculations -- 28. The raw data.
Summary	A polytope exchange transformation is a (discontinuous) map from a polytope to itself that is a translation wherever it is defined. The 1-dimensional examples, interval exchange transformations, have been studied fruitfully for many years and have deep connections to other areas of mathematics, such as Teichmüller theory. This book introduces a general method for constructing polytope exchange transformations in higher dimensions and then studies the simplest example of the construction in detail. The simplest case is a 1-parameter family of polygon exchange transformations that turns out to be closely related to outer billiards on semi-regular octagons. The 1-parameter family admits a complete renormalization scheme, and this structure allows for a fairly complete analysis both of the system and of outer billiards on semi-regular octagons. The material in this book was discovered through computer experimentation. On the other hand, the proofs are traditional, except for a few rigorous computer-assisted calculations.
Bibliography note	Includes bibliographical references (pages 211-212).
LCCN	2014006823
ISBN	9781470415228 (alk. paper)
ISBN	1470415224 (alk. paper)

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The octagonal PETs / Richard Evan Schwartz.

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