## Project CEMAPRE internal

Title | Lyapunov exponents of randomly perturbed billiards |

Participants | Gianluigi Del Magno, José Pedro Gaivão (Principal Investigator), João Lopes Dias |

Summary | This project deals with Lyapunov exponents, which is a quantity that measures the complexity of dynamical systems. More precisely, Lyapunov exponents quantify the rate of separation of nearby orbits, and positive Lyapunov exponents indicates that the system has sensitive dependence on initial conditions, a phenomenon also known as the butterfly effect. Establishing positivity of Lyapunov exponents of a given dynamical system is a highly non-trivial task. In most applications, dynamical systems are subject to random external perturbations. If the unperturbed system has some hyperbolicity, then random perturbations with absolutely continuous probability transitions are expected to produce positive Lyapunov exponents. A notable class of dynamical systems are the mathematical billiards. When the billiard table is strictly convex and non-integrable, it is long standing open problem to prove the positivity of the top Lyapunov exponent. |