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On non-archimedean dynamical systems
Thesis (MSc) -- University of Stellenbosch, 2000.
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2012
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| _version_ | 1867613990823133184 |
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| access_status_str | Open Access |
| author | Joyner, Sheldon T |
| author2 | Green, B. W. |
| author_browse | Green, B. W. Joyner, Sheldon T |
| author_facet | Green, B. W. Joyner, Sheldon T |
| author_sort | Joyner, Sheldon T |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc) -- University of Stellenbosch, 2000. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/51861 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:44:55.985Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2012 |
| publishDateRange | 2012 |
| publishDateSort | 2012 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/51861 On non-archimedean dynamical systems Joyner, Sheldon T Green, B. W. Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences (applied, computer, mathematics). Differentiable dynamical systems Sets JULIA set FATOU set Dissertations--Mathematics Theses -- Mathematics Thesis (MSc) -- University of Stellenbosch, 2000. ENGLISH ABSTRACT: A discrete dynamical system is a pair (X, cf;) comprising a non-empty set X and a map cf; : X ---+ X. A study is made of the effect of repeated application of cf; on X, whereby points and subsets of X are classified according to their behaviour under iteration. These subsets include the JULIA and FATOU sets of the map and the sets of periodic and preperiodic points, and many interesting questions arise in the study of their properties. Such questions have been extensively studied in the case of complex dynamics, but much recent work has focussed on non-archimedean dynamical systems, when X is projective space over some field equipped with a non-archimedean metric. This work has uncovered many parallels to complex dynamics alongside more striking differences. In this thesis, various aspects of the theory of non-archimedean dynamics are presented, with particular reference to JULIA and FATOU sets and the relationship between good reduction of a map and the empty JULIA set. We also discuss questions of the finiteness of the sets of periodic points in special contexts. AFRIKAANSE OPSOMMING: 'n Paar (X, <jJ) bestaande uit 'n nie-leë versameling X tesame met 'n afbeelding <jJ: X -+ X vorm 'n diskrete dinamiese sisteem. In die bestudering van so 'n sisteem lê die klem op die uitwerking op elemente van X van herhaalde toepassing van <jJ op die versameling. Elemente en subversamelings van X word geklasifiseer volgens dinamiese kriteria en op hierdie wyse ontstaan die JULIA en FATOU versamelings van die afbeelding en die versamelings van periodiese en preperiodiese punte. Interessante vrae oor die eienskappe van hierdie versamelings kom na vore. In die geval van komplekse dinamika is sulke vrae reeds deeglik bestudeer, maar onlangse werk is op nie-archimediese dinamiese sisteme gedoen, waar X 'n projektiewe ruimte is oor 'n liggaam wat met 'n nie-archimediese norm toegerus is. Hierdie werk het baie ooreenkomste maar ook treffende verskille met die komplekse dinamika uitgewys. In hierdie tesis word daar ondersoek oor verskeie aspekte van die teorie van nie-archimediese dinamika ingestel, in besonder met betrekking tot die JULIA en FATOU versamelings en die verband tussen goeie reduksie van 'n afbeelding en die leë JULIA versameling. Vrae oor die eindigheid van versamelings van periodiese punte in spesiale kontekste word ook aangebied. 2012-08-27T11:34:43Z 2012-08-27T11:34:43Z 2000-12 Thesis http://hdl.handle.net/10019.1/51861 en_ZA Stellenbosch University 101 p. application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Differentiable dynamical systems Sets JULIA set FATOU set Dissertations--Mathematics Theses -- Mathematics Joyner, Sheldon T On non-archimedean dynamical systems |
| title | On non-archimedean dynamical systems |
| title_full | On non-archimedean dynamical systems |
| title_fullStr | On non-archimedean dynamical systems |
| title_full_unstemmed | On non-archimedean dynamical systems |
| title_short | On non-archimedean dynamical systems |
| title_sort | on non archimedean dynamical systems |
| topic | Differentiable dynamical systems Sets JULIA set FATOU set Dissertations--Mathematics Theses -- Mathematics |
| url | http://hdl.handle.net/10019.1/51861 |
| work_keys_str_mv | AT joynersheldont onnonarchimedeandynamicalsystems |