Formal-group isogeny conjecture for stable p-adic dynamical systems

Let $\mathcal{D}$ be a stable $p$-adic dynamical system, and let $f$ and $u$ be, respectively, stable noninvertible and invertible power series in $\mathcal{D}$. A formal group $F$ with coefficients in $\mathcal{O}_K$ has endomorphisms $f_F$ and $u_F$. A nonzero power series $h$ is an isogeny from $f_F$ to $f$ when

$$f \circ h=h \circ f_F$$

and

$$u \circ h=h \circ u_F.$$

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 that

$$f \circ h=h \circ f_F,\qquad u \circ h=h \circ u_F.$$

The conjecture predicts that every such stable $p$-adic dynamical system is simultaneously intertwined with endomorphisms of a formal group. Its resolution would provide a formal-group description of the dynamics and clarify the structure of stable systems.