On a Slice of the Cubic 2-adic Mandelbrot Set
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| Format: | Preprint |
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2024
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| _version_ | 1866910299854995456 |
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| author | Anderson, Jacqueline Stacy, Emerald Tobin, Bella |
| author_facet | Anderson, Jacqueline Stacy, Emerald Tobin, Bella |
| contents | Consider the one-parameter family of cubic polynomials defined by $f_t(z) =-\frac 32 t(-2z^3+3z^2)+1, t \in \mathbb{C}_2$. This family corresponds to a slice of the parameter space of cubic polynomials in $\mathbb{C}_2[z]$. We investigate which parameters in this family belong to the cubic $2$-adic Mandelbrot set, a $p$-adic analog of the classical Mandelbrot set. When $t=1$, $f_t(z)$ is post-critically finite with a strictly preperiodic critical orbit. We establish that this is a non-isolated boundary point on the cubic $2$-adic Mandelbrot set and show asymptotic self-similarity of the Mandelbrot set near this point. Subsequently, we investigate the Julia set for polynomial on the boundary and demonstrate a similarity between the Mandelbrot set at this point and the Julia set, similar to what is seen in the classical complex case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_09394 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a Slice of the Cubic 2-adic Mandelbrot Set Anderson, Jacqueline Stacy, Emerald Tobin, Bella Dynamical Systems Number Theory 11C08, 37P05, 37P40 Consider the one-parameter family of cubic polynomials defined by $f_t(z) =-\frac 32 t(-2z^3+3z^2)+1, t \in \mathbb{C}_2$. This family corresponds to a slice of the parameter space of cubic polynomials in $\mathbb{C}_2[z]$. We investigate which parameters in this family belong to the cubic $2$-adic Mandelbrot set, a $p$-adic analog of the classical Mandelbrot set. When $t=1$, $f_t(z)$ is post-critically finite with a strictly preperiodic critical orbit. We establish that this is a non-isolated boundary point on the cubic $2$-adic Mandelbrot set and show asymptotic self-similarity of the Mandelbrot set near this point. Subsequently, we investigate the Julia set for polynomial on the boundary and demonstrate a similarity between the Mandelbrot set at this point and the Julia set, similar to what is seen in the classical complex case. |
| title | On a Slice of the Cubic 2-adic Mandelbrot Set |
| topic | Dynamical Systems Number Theory 11C08, 37P05, 37P40 |
| url | https://arxiv.org/abs/2401.09394 |