Expansions of quadratic maps in prime fields

Abstract:

Let $f(x)=ax^2+bx+c\in \mathbb Z[x]$ be a quadratic polynomial with $a\not \equiv 0$ mod $p$. Take $z\in \mathbb F_p$ and let $\mathcal O_z=\{f_i(z)\}_{i\in \mathbb Z^+}$ be the orbit of $z$ under $f$, where $f_i(z)=f(f_{i-1}(z))$ and $f_0(z)=z$. For $M< |\mathcal O_z|$, we study the diameter of the partial orbit $\mathcal O_M=\{z, f(z), f_2(z),\dots , f_{M-1}(z)\}$ and prove that there exists $c_1>0$ such that \[ \operatorname {diam} \mathcal O_M \gtrsim \min \bigg \{ Mp^{\;{c_1}}, \frac 1{\log p} M^{\frac 45} p^{\frac 15}, M^{\;\frac 1{13}\log \log M}\bigg \}. \] For a complete orbit $\mathcal C$, we prove that \[ \operatorname {diam} \mathcal C \gtrsim \min \{p^{\;5c_1}, e^{ \;T/4}\;\},\] where $T$ is the period of the orbit.