Modeli mnogodelčne dinamike v obliki klasičnih in kvantnih vezij

Treating the time evolution of many-body systems in discrete time steps is very useful in both classical and quantum physics, simplifying exact and numerical calculations of different observables. In this thesis, we study different classical and quantum discrete-time models. In the first part, we focus on classical systems through the lens of cellular automata which, despite being simple in construction, exhibit a multitude of differing integrable and chaotic behaviors. We study a specific subset of momentum-conserving parity check automata (MCPCA), which are cellular automata defined on bipartite graphs that conserve momentum and parity along their interaction vertices. We show that these automata describe systems with non-ergodic behavior in the spectra of their autocorrelation functions for different types of lattices. Studying their conserved charges, we demonstrate that the information spreading in such systems undergoes a phase transition between a localized and a delocalized phase. Additionally, we investigate the transport properties of the conserved charges in a specific sector of charge configuration of the MCPCA, we observe that the transport is anomalous in this sector. Moreover, we provide a classification of all possible two-states 3-to-3 cellular automata rules defined on the honeycomb lattice based on their dynamics and spectral properties. In the final part of the thesis, we focus on the quantum dual-unitary gates, which are a special kind of quantum gates that preserve their unitarity condition under space-time exchange. We give exact results on the dynamical correlation functions of dual-unitary circuits defined on random geometries that we dubbed Mikado circuits. We show different random geometries realizations and the exact results for the first two moments of the averaged correlation functions.