Dynamics of Topologically Generic Homeomorphisms

by Akin, Ethan; Kennedy, Judy A.; Hurley, Mike

• ISBN13: 9780821833384
• ISBN10: 0821833383
• Format: Paperback
• Copyright: 2003-07-01
• Publisher: Amer Mathematical Society

Summary

Introduction Attractors and chain recurrence Periodic decompositions and adding machines Semicontinuity and homogeneity Crushing arguments Topological horseshoes Generic homeomorphisms Almost equicontinuity Cantor sets The circle Crushing the chain recurrent set Generic homeomorphisms on manifolds Generic mappings on manifolds Bibliography Index

Table of Contents

Introduction	1	(1)
Overview	1	(1)
Description of results	2	(3)
Brief remarks about techniques	5	(1)
Comparison with the smooth case	6	(1)
Standing notation	7	(1)
Road map	8	(1)
Comments	8	(3)
Attractors and Chain Recurrence	11	(10)
Chain recurrence	11	(2)
Attractors	13	(2)
Attractor-repellor pairs	15	(1)
Initial and terminal chain components	16	(1)
The space(s) of chain components	17	(1)
Summary	18	(3)
Periodic Decompositions and Adding Machines	21	(10)
Periodic decompositions	21	(1)
The sets II∞ and II∞,∞	22	(1)
Adding machines	23	(2)
Decompositions of u type	25	(3)
Periodicity conditions	28	(3)
Semicontinuity and Homogeneity	31	(8)
Semicontinuity	31	(1)
Prolongation	32	(1)
The automorphism group	33	(2)
Homogeneity conditions	35	(1)
Showing Ω(ƒ) = C(ƒ)	36	(3)
Crushing Arguments	39	(14)
Sponges	39	(5)
Crushing	44	(9)
Topological Horseshoes	53	(12)
Connected successions of subsets	53	(1)
Topological horseshoes	54	(3)
Perturbing to a horseshoe	57	(8)
Generic Homeomorphisms	65	(14)
The classes H1 and H1,s	65	(5)
The class H,s	70	(3)
Dynamic isolation	73	(1)
Attractor boundaries are quasi-attractors	73	(2)
Shift extensions and the class H4	75	(4)
Almost Equicontinuity	79	(6)
Chain continuity	81	(4)
Cantor Sets	85	(12)
Aperiodicity and the class H5	85	(1)
The class H6	86	(3)
Rohlin property	89	(1)
The class H3,c	90	(7)
The Circle	97	(10)
Background	97	(2)
The class H1 on S1	99	(4)
Relative Rohlin property	103	(4)
Crushing the Chain Recurrent Set	107	(8)
Generic Homeomorphisms on Manifolds	115	(6)
The class H8	115	(1)
The class Hman	116	(1)
Anosov homeomorphisms	116	(5)
Generic Mappings on Manifolds	121	(4)
Bibliography	125	(4)
Index	129