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.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/51861 |
| Date | 12 1900 |
| Creators | Joyner, Sheldon T |
| Contributors | Green, B. W., Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences (applied, computer, mathematics). |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | Unknown |
| Type | Thesis |
| Format | 101 p. |
| Rights | Stellenbosch University |
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