  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.
