ETD

• Cerbelli-Giona map
• C^1-regularization
• lyapunov exponents
• twist maps

Lyapunov exponents constitute a fundamental tool in the study of chaos in dynamical systems. In this thesis, we provide sufficient conditions, numerically satisfied, such that for a given dynamical system subject to small $C^0$ perturbations, positive Lyapunov exponents exist on a set of positive Lebesgue measure. First, we consider a family of area-preserving, piecewise linear twist maps of the torus, including the Cerbelli-Giona map. We show that these maps belong to the class of pseudo-Anosov systems and exhibit positive Lyapunov exponents almost everywhere. Subsequently, through a regularization procedure, we obtain $C^1$ maps and we investigate their dynamical properties. Focusing our study on the perturbation of the Cerbelli-Giona map, we provide sufficient conditions based on the geometry of the stable and unstable manifolds associated with a hyperbolic periodic point to ensure that the perturbed map is ergodic and transitive. These results are then used to demonstrate the positivity of Lyapunov exponents on a set of positive Lebesgue measure. We conclude by presenting numerical simulations that support the theoretical analysis.