Substitution maps in the Robba ring (Q2080523)

Let $p$ be a prime number, and $K$ is a finite extension of $Q_{p}$. Let $O_{K}$ denote the integers of $K$, let $m_{K}$ be the maximal ideal of $O_{K}$, let $k$ denotes the residue field of $O_{K}$, and let $\pi$ be a uniformizer of $O_{K}$. $\Vert.\Vert$ is a $p$-adic norm on $K$. In $p$-adic Hodge theory, the theory of $p$-adic differential equations, and the theory of $p$-adic dynamical systems, several rings of power series with coefficients in $K$ such as Robba ring occur. These rings are often endowed with a substitution map $\varphi$ of the form $\varphi:f(X)\mapsto f(s(X))$, where $s(X)$ is either a Frobenius lift or a more general power series. The author of the paper under review asks several questions about substitution maps in the Robba ring. His questions are motivated by $p$-adic Hodge theory and the theory of $p$-adic dynamical systems. In the paper under review by providing answers to his questions under special conditions he generalizes results of Kedlaya, Colmez, and others. He presents two conjectures and partially answers them in the paper. It is a good work in $p$-adic mathematical subjects.