Dynamics of Rational Functions and Wandering Domains

Date

2023-5-25

Author

Saraçlar, İpek

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As Sullivan proved in 1985, there is no rational map whose Fatou components possess a wandering domain in one dimension, and in the proof quasi-conformal mappings were used. After that, it is expected that there should be a map, but not in one dimension, and that not every component of the Fatou set of this rational map is eventually periodic. This thesis will be a review of which maps have been built over the years to see the dynamics of wandering components. Chronologically, polynomial skew-product mappings in two dimensions were constructed using parabolic implosion techniques, and it’s seen that they have wandering domains [3]. In 2004, Lilov proved that, near an invariant super-attracting fiber the wandering Fatou components of polynomial skew- products cannot exist [2]. Then, the examples of mappings with wandering domains have been given explicitly in [3] and [12]. The concept of strongly-attracting fiber is defined in the paper of Ji(2018) [8] concluding that there are no wandering Fatou components near strongly-attracting fiber with the given conditions.

Subject Keywords

iteration, Fatou-Julia sets, wandering domain, polynomial skew-product

URI

https://hdl.handle.net/11511/104402

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Graduate School of Natural and Applied Sciences, Thesis