In this paper, we study dynamics of maps on quasi-graphs characterizing their invariant measures. particular, prove that every measure quasi-graph map with zero topological entropy has discrete spectrum. Additionally, obtain an analog Llibre-Misiurewicz's result relating positive existence horseshoes. We also dendrites and show if a continuous dendrite, whose set all endpoints is closed only fi...

Discrete time crystals are periodically driven systems that display spontaneous symmetry breaking of translation invariance in the form indefinite subharmonic oscillations. We introduce a thermodynamically consistent model for discrete crystal and analyze it using framework stochastic thermodynamics. In particular, we evaluate rate energy dissipation this many-body system interacting noisy osci...