In the theory of non-Archimedean dynamical systems, we consider the projective line over algebraically closed, complete, and non-Archimedean normed fields such as the complex p-adic field C_p with the p-adic norm as the spaces of dynamical systems.

The theory is relatively new and mostly developed in this century so there are a lot of assignments to accomplish in order to understand the dynamics. One of the assignments is about the stability of the dynamical systems, which states that if two given dynamical systems are "close enough", then there exists a conjugation between those two dynamical systems.

This talk will provide a brief review of the theory of non-Archimedean dynamical systems and we will see the main result, the stability of Julia sets of immediately expanding polynomial maps over non-Archimedean field, of this talk. As an example of the main result, we will also see the dynamics of the families of Mandelbrot maps {z^d + c | c in C_p}, which are an analogue of the families of maps that induce the Mandelbrot sets in complex dynamics.