38 problems

No-wandering-domains conjecture. Such a rational map has no wandering domains in $\mathbb{P}^1_{\mathbb{C}_p}$.

Let $\mathcal{O}$ be the ring of integers of a finite extension of $\mathbb{Q}_p$. Let $f$ and $u$ be, respectively, a noninvertible series and a nontorsion invertible series defin…

Let $\phi$ be the rational map under consideration, with $J_{\mathbb{C}_p}(\phi)$ its $\mathbb{C}_p$-Julia set and $J_{\overline{\mathbb{Q}}_p}(\phi)$ the subset associated with…

Let $g$ be a polynomial of degree $d\geq 2$ with coefficients in a number field $K$, and let $y_0\in K$. Define the Newton map … and, for each $n\geq 0$, set $y_{n+1}=N(y_n)$. Assu…

Autoconjugacy Conjecture.

Rationality conjecture for $Q_a$. The map $Q_a$ maps the rationals with odd denominators to rational numbers. The bibliography states that this conjecture is true for $a=\pm1$, fal…

Conjecture de pré-périodicité des composantes. Toutes les composantes de l'ensemble de Fatou sont pré-périodiques.

Lubin's conjecture. Under these hypotheses,

The Cantor-type basin conjecture. If $R$ has an attracting cycle whose immediate basin is of Cantor type and contains no critical point of $R$, then $R$ has infinitely many attract…

Minimality conjecture. Outside the $p^{-1}$-neighborhoods of these periodic points, minimality holds over the $p$-adic integer points; equivalently, the relevant automorphism dynam…

Let $\mathbb{Z}_2$ be the ring of $2$-adic integers, let $\overline{T}\colon\mathbb{Q}_2\to\mathbb{Q}_2$ be given by … where at each iterate $v$ is the residue modulo $2$ of the cu…

Let $K$ be a finite extension of $\mathbf{Q}_p$, with ring of integers $\mathcal{O}_K$ and maximal ideal $\mathfrak{m}_K$. Let $P,U\in T\cdot\mathcal{O}_K[[T]]$ satisfy … with…

Let $p$ be a prime, let $d\geq 1$, and write $X=(x_1,\ldots,x_d)$. Let $\operatorname{Preper}(u)$ denote the preperiodic points of $u$, and let $\Lambda(f)$ denote the zero set of…

Let $p$ be a prime, let $d\geq 1$, and write $X=(x_1,\ldots,x_d)$. For a power series $h\in\mathbb{Z}_p[[X]]^d$, let $\Lambda(h)$ denote its zero set. Commuting noninvertible forma…

Salerno and Silverman's conjecture. The radius of convergence of $\phi^{-1}(x)$ in $\mathbb{C}_{p}$ is

Julia-set equality conjecture. For every rational map $\phi\in K(z)$,

Positive-characteristic $p$-adic interpolation conjecture. Under these hypotheses, the $f$-periodic points are not dense in $(K^{\circ})^r$ with respect to the $t$-adic topology.

Formal-group isogeny conjecture. There exists a formal group $F$ with coefficients in $\mathcal{O}_K$, two endomorphisms $f_F$ and $u_F$ of $F$, and a nonzero power series $h$ such…

Growing-cycle conjecture. If $m\equiv \pm 2\pmod{10}$, then $f$ has $5^{t-1}$ growing cycles of length $4$ at every level $\geq t+1$; if $m\equiv \pm 7\pmod{10}$, then $f$ has…

Let $f(x)$ be a polynomial over a finite extension $K/{\mathbb Q}_p$ of degree not divisible by $p$, assume that $f$ has good reduction, and let $alpha$ be a point not in the Julia…

Let $p$ be a prime and let $r\ge0$. Define … and let … be the associated Böttcher coordinate. Asymptotic valuation and radius conjecture. For $k\equiv0\pmod p$, … Moreover, the rad…

Let $p$ be a prime and let … Write the Böttcher coordinate of $\varphi$ as … where the coefficients $a_k$ are integers. Coefficient congruence conjecture. The following congruences…

Let $F$ be a field of positive characteristic, let $\varphi:\mathbb{P}^{1}_{F}\to\mathbb{P}^{1}_{F}$, and let $0$ be a fixed point with multiplier $\lambda$. Isolation conjecture.…

Let $\mathfrak{f}$ and $\mathfrak{u}$ be a non-invertible and a non-torsion invertible series, respectively, defined over the ring of integers $\mathcal{O}$ of a finite extension o…

Let $f$ be an analytic map over an ultrametric field of positive characteristic, and let $\zeta_0$ be a parabolic periodic point, meaning that for its minimal period $n$, the multi…