A Connection between Analytic and Nonanalytic Singular Perturbations of the Quadratic Map
2017-05
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A Connection between Analytic and Nonanalytic Singular Perturbations of the Quadratic Map
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2017-05
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Abstract
To explore the connection between the analytic and the nonanalytic complex dy-
namics, this paper studied a special case of the singularly perturbed quadratic map:
β β
ƒβ‚t (z) = z2 + t — + (1 – t) —
z2 – z2
which can transit from nonanalytic to analytic by varying t from 0 to 1. Other variables
involved in this map are the dynamic variable z ϵ C and the main parameter β ϵ R.
Our investigation shows that this transition map has much richer behaviors than the end
point cases. The parameter space can be no longer subdivided into only four or three
regions as shown in the studies by Devaney and Bozyk respectively. Correspondingly,
in the dynamic plane, there also appear several new intermediate cases among which we
identified two transitions: a connected case where the filled Julia set is connected and
a disconnected case where the filled Julia set consists of infinitely many components.
This paper also pointed out that ƒβ‚t (z) is semiconjugate to the four symbols shift map
for the disconnected case. This semiconjugacy provides a way to largely understand the
dynamical behaviors for the non escape points. Further study shows that the critical
set plays an important role in the construction of the filled Julia set. Therefore, the
transition of the critical set and its image set are also discussed in this paper. At the
end, we presented two sets of conjectures for the bounded critical orbits and the escape
critical orbits for future study.
Description
University of Minnesota M.S. thesis. May 2017. Major: Mathematics and Statistics. Advisor: Bruce Peckham. I computer file (PDF); vi, 51 pages, appendix A, Ill.
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Liu, Yujiong. (2017). A Connection between Analytic and Nonanalytic Singular Perturbations of the Quadratic Map. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/189081.
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