Distribution of preperiodic points in one-parameter families of rational maps
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866912545796784128 |
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| author | Olechnowicz, Matt |
| author_facet | Olechnowicz, Matt |
| contents | Let $f_t$ be a one-parameter family of rational maps defined over a number field $K$. We show that for all $t$ outside of a set of natural density zero, every $K$-rational preperiodic point of $f_t$ is the specialization of some $K(T)$-rational preperiodic point of $f$. Assuming a weak form of the Uniform Boundedness Conjecture, we also calculate the average number of $K$-rational preperiodic points of $f$, giving some examples where this holds unconditionally. To illustrate the theory, we give new estimates on the average number of preperiodic points for the quadratic family $f_t(z) = z^2 + t$ over the field of rational numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_00933 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Distribution of preperiodic points in one-parameter families of rational maps Olechnowicz, Matt Number Theory Dynamical Systems Let $f_t$ be a one-parameter family of rational maps defined over a number field $K$. We show that for all $t$ outside of a set of natural density zero, every $K$-rational preperiodic point of $f_t$ is the specialization of some $K(T)$-rational preperiodic point of $f$. Assuming a weak form of the Uniform Boundedness Conjecture, we also calculate the average number of $K$-rational preperiodic points of $f$, giving some examples where this holds unconditionally. To illustrate the theory, we give new estimates on the average number of preperiodic points for the quadratic family $f_t(z) = z^2 + t$ over the field of rational numbers. |
| title | Distribution of preperiodic points in one-parameter families of rational maps |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2306.00933 |