Lyapunov exponents are natural observables to quantify the chaoticity of a trajectory. They thus appear as good candidates to discriminate between different dynamical regimes, allowing to study phenomena such as the onset of turbulence—which goes hand in hand with the emergence of chaotic trajectories in an otherwise regular flow—or the glass transition—which can be seen as a transition from diffusive dynamics to an arrested, frozen-in, and ergodicity-breaking regime.

The present thesis strives to apply the thermodynamic formalism of Sinai, Ruelle and Bowen—which transposes in trajectory space the language of equilibrium statistical physics—to fluctuations of Lyapunov exponents in spatially extended systems, for which only few results are available.

We begin by presenting a numerical method to sample trajectories of atypical chaoticity in spatially extended systems, hence revealing their various dynamical structures. We also exhibit how this algorithm can be used to measure the dynamical free energy, opening the way for the study of dynamical phase transitions resulting from the possible coexistence of these structures. This method is in particular applied to the Fermi-Pasta-Ulam-Tsingou (FPU) chain of anharmonic oscillators.

Next, we show how fluctuations of the largest Lyapunov exponent in systems of interacting particles with underlying diffusive dynamics can be analytically characterized. Carrying out this program allows us to establish interesting connections with damage spreading and reaction-diffusion processes.

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