## Project Details

### Description

(1) We consider a 1-parameter family $H_{t} : \AA^{2} \to \AA^{2}$ of Henon maps, where $H_{t}(x, y) = (y + f_{t}(x), x)$ and $f_{t}(x) \in K[x, t]$ with the degree (in $x$) of $f_{t}(x)$ being at least 2. Let $\Sigma(\bfP)$ denotes the set of parameters $t\in \Kbar$ where the specialized point $P_{t} \in \AA^{2}(K[t])$ is periodic for the specialized H\'enon map $H_{t}.$ We're interested in conditions satisfied by $\bfP$ and $H$ under which the set of periodic parameters $\Sigma(\bfP)$ is an infinite set. (2) Let $K$ be the function field of a smooth, irreducible curve defined over $\Qbar$. Let $f\in K[x]$ be of the form $f(x)=x^p+c$ for $p$ prime, and let $\beta\in \Kbar$. For all $n\in\mathbb{N}\cup\{\infty\}$, the Galois groups $G_n(\beta)=\Gal(K(f^{-n}(\beta))/K(\beta))$ embed into $[C_p]^n$, the $n$-fold wreath product of the cyclic group $C_p$. We show that if $f$ is not isotrivial, then $[[C_p]^\infty:G_\infty(\beta)]<\infty$ unless $\beta$ is postcritical or periodic. We are also able to prove that if $f_1(x)=x^p+c_1$ and $f_2(x)=x^p+c_2$ are two such polynomials, then the fields $\bigcup_{n=1}^\infty K(f_1^{-n}(\beta))$ and $\bigcup_{n=1}^\infty K(f_2^{-n}(\beta))$ are disjoint over a finite extension of $K$.

Status | Finished |
---|---|

Effective start/end date | 2017/08/01 → 2019/07/31 |

### Keywords

- arithmetic dynamics
- unlikely intersection questions
- 1-parameter family
- Hénon maps
- periodic points
- dynamically related
- arboreal Galois representations
- iterated Galois groups

## Fingerprint

Explore the research topics touched on by this project. These labels are generated based on the underlying awards/grants. Together they form a unique fingerprint.