تقرير
Totally real algebraic numbers in generalized Mandelbrot set
| العنوان: | Totally real algebraic numbers in generalized Mandelbrot set |
|---|---|
| المؤلفون: | Hare, Kevin G., Noytaptim, Chatchai |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Dynamical Systems, Mathematics - Number Theory, 37F46, 11R80, 37P05 |
| الوصف: | In this article, we study some potential theoretical and topological aspects of the generalized Mandelbrot set introduced by Baker and DeMarco. For $\alpha$ real, we study the set of all totally real algebraic parameters $c$ such that $\alpha$ is preperiodic under the iteration of the one-parameter family $f_c(x) = x^2 + c$. We show that when $|\alpha| < 2$ and rational then the set of totally real algebraic parameters $c$ with this property is finite, whereas if $|\alpha| \geq 2$ and rational then this set is countably infinite. As an unexpected consequence of this study, we also show that when $|\alpha| \geq 2$ then parameters $c$ such that $\alpha$ is $f_c$-periodic are necessarily real. As a special case, we classify all totally real algebraic integers $c$ such that $\alpha = \pm1$ is preperiodic. Comment: 13 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2405.10395 |
| رقم الأكسشن: | edsarx.2405.10395 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |