Appears in collections : Jean-Morlet Chair 2021 - Conference: Faces of Singularity Theory / Chaire Jean-Morlet 2021 - Conférence : Visages de la théorie des singularités, Logarithmic and non-archimedean methods in Singularity Theory - Thematic Month Week 1 / Méthodes logarithmiques et non-archimédiennes en théorie des singularités - Mois thématique semaine 1

We compare the topological Milnor fibration and the motivic Milnor fibreby introducing a common extension : the complete Milnor fibration. This extension is constructed using either logarithmic geometry or an oriented (multi)graph construction, for a complex regular function with only normal crossings. The comparison uses quotients by the action of the group of positive real numbers. We study moreover how this model changes under blowings-up. Joint work with J.-B. Campesato and A. Parusinski.