{"@context":"http://iiif.io/api/presentation/2/context.json","@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/manifest.json","@type":"sc:Manifest","label":"On the route to chaos for two-dimensional modestly area-contracting analytic maps","metadata":[{"label":"dc.description.sponsorship","value":"This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree."},{"label":"dc.format","value":"Monograph"},{"label":"dc.format.medium","value":"Electronic Resource"},{"label":"dc.identifier.uri","value":"http://hdl.handle.net/11401/76390"},{"label":"dc.language.iso","value":"en_US"},{"label":"dc.publisher","value":"The Graduate School, Stony Brook University: Stony Brook, NY."},{"label":"dcterms.abstract","value":"It has long been conjectured that the two-dimensional dissipative maps, like their one-dimensional counterparts, experience the period-doubling cascade to chaos. In [CEK], Collet, Eckmann and Koch proved this conjecture for highly dissipative families. In this dissertation, we introduce  the notion of nested systems that generalizes Henon maps, and construct two operators acting on the space of nested systems based on the idea of the return maps. We use the two operators to show that if a dissipative nested system satisfies some apriori bounds, then they can be replaced  with simpler but dynamically equivalent nested systems. We prove that the total number of applications of the operators is bounded by the number of periodic points. When the procedure of the applications stops, we obtain ``little Henon'' maps whose dynamics are well-understood. We then show that if the first nested system of a family  contains finitely many periodic points, then the family only  experiences either saddle-node or periodic-doubling bifurcations. We conclude that if there are sufficiently many periodic points, then moderate dissipative nested systems can be transformed into highly dissipative ones so that the result of [CEK] applies."},{"label":"dcterms.available","value":"2017-09-20T16:50:09Z"},{"label":"dcterms.contributor","value":"Winckler, Bjorn"},{"label":"dcterms.creator","value":"Chi, Ying"},{"label":"dcterms.dateAccepted","value":"2017-09-20T16:50:09Z"},{"label":"dcterms.dateSubmitted","value":"2017-09-20T16:50:09Z"},{"label":"dcterms.description","value":"Department of Mathematics."},{"label":"dcterms.extent","value":"101 pg."},{"label":"dcterms.format","value":"Application/PDF"},{"label":"dcterms.identifier","value":"http://hdl.handle.net/11401/76390"},{"label":"dcterms.issued","value":"2015-12-01"},{"label":"dcterms.language","value":"en_US"},{"label":"dcterms.provenance","value":"Made available in DSpace on 2017-09-20T16:50:09Z (GMT). No. of bitstreams: 1\nChi_grad.sunysb_0771E_12586.pdf: 815618 bytes, checksum: 0328ba73b384593d9f650695fc4b9482 (MD5)\n  Previous issue date:    1"},{"label":"dcterms.publisher","value":"The Graduate School, Stony Brook University: Stony Brook, NY."},{"label":"dcterms.subject","value":"Mathematics"},{"label":"dcterms.title","value":"On the route to chaos for two-dimensional modestly area-contracting analytic maps"},{"label":"dcterms.type","value":"Dissertation"},{"label":"dc.type","value":"Dissertation"}],"description":"This manifest was generated dynamically","viewingDirection":"left-to-right","sequences":[{"@type":"sc:Sequence","canvases":[{"@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/canvas/page-1.json","@type":"sc:Canvas","label":"Page 1","height":1650,"width":1275,"images":[{"@type":"oa:Annotation","motivation":"sc:painting","resource":{"@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/15%2F07%2F67%2F150767966063731907754807524126327791142/full/full/0/default.jpg","@type":"dctypes:Image","format":"image/jpeg","height":1650,"width":1275,"service":{"@context":"http://iiif.io/api/image/2/context.json","@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/15%2F07%2F67%2F150767966063731907754807524126327791142","profile":"http://iiif.io/api/image/2/level2.json"}},"on":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/canvas/page-1.json"}]}]}]}