to be equipped by a non-Archimedean norm (to be defined later). In this case, there is an interesting construction existing over this field which is called a Berkovich space. We will see that this space has a natural tree structure and that the map $F_t$ acts naturally on this tree which provides a dynamical system giving information about how the family $[f_t]\in\rat_d$ diverges when $t\neq 0$.

[**Outline.**]{} This paper will try to be pedagogic. Hence will first make an exposition of the comparison of these two notions emphasizing the ideas and forgetting a little about the Berkovich general formalism that will be introduced in Section \[Chap7\]. As in order to do dynamics we first need a space, then a map and finally a way to iterate, the structure of this paper will be the sequel: spaces Section \[Chap2\], maps Section \[Chap3\] and dynamics Section \[Chap4\]. Then in Section \[Chap5\] we will compare the approaches underlying advantages and disadvantages of these two ones. Finally we will have Section \[Chap7\] as already explained.

[**Acknowledgments.**]{} This paper follows from discussions with Jan Kiwi during my PhD. I want to thank also Charles Favre, Mattias Jonsson and Juan Rivera-Letelier for discussion that allowed me 