Dynamical phase transitions in glasses induced by the ruggedness of the free energy landscape

We propose damage spreading (DS) as a tool to investigate the topological features related to the ruggedness of the free energy landscape. We argue that DS measures the positiveness of the largest Lyapunov exponent associated to the basins of attraction visited by the system during its dynamical evolution. We discuss recent results obtained in the framework of mode-coupling theory and comment how to extend them to the study of realistic glasses. Preliminary results are presented for purely repulsive soft-sphere glasses.