Unlikely intersection problems for restricted lifts of p-th power

I define a morphism on $\mathbb{C}_p$ called a lift of $p$-th power if its natural restriction to the residue field of $\mathbb{C}_p$ is a $p$-th power of some morphism. This definition generalizes from the lift of Frobenius. In this paper the theory of perfectoid spaces is applied on the morphism to the Manin-Mumford conjecutre, the Mordell-Lang conjecture and the Tate-Volch conjecture.