Title: Rare events, time crystals and symmetry-breaking dynamical phase transitions

Abstract: In this PhD thesis, I investigate the properties of symmetry-breaking dynamical phase transitions that manifest in the fluctuations of time-integrated observables within classical systems. In particular, I analyze how these phase transitions impose stringent constraints on the structure of the eigenvectors of the system dynamical generator of the dynamics. Additionally, I identify a dynamical phase transition to a time-crystal phase in a model of driven-diffusive lattice gas. The study of this transition then allows the identification of the "packing-field" mechanism responsible for its emergence. This mechanism is then exploited to propose new transport models displaying time-crystal behavior.