Finding the Effective Dynamics to Make Rare Events Typical in Chaotic Maps

Dynamical fluctuations or rare events associated with atypical trajectories in chaotic maps due to specific initial conditions can crucially determine their fate, as the may lead to stability islands or regions in phase space otherwise displaying unusual behavior. Yet, finding such initial conditions is a daunting task precisely because of the chaotic nature of the system. In this Letter, we circumvent this problem by proposing a framework for finding an effective topologically conjugate map whose typical trajectories correspond to atypical ones of the original map. This is illustrated by means of examples which focus on counterbalancing the instability of fixed points and periodic orbits, as well as on the characterization of a dynamical phase transition involving the finite-time Lyapunov exponent. The procedure parallels that of the application of the generalized Doob transform in the stochastic dynamics of Markov chains, diffusive processes, and open quantum systems, which in each case results in a new process having the prescribed statistics in its stationary state. This Letter thus brings chaotic maps into the growing family of systems whose rare fluctuations-sustaining prescribed statistics of dynamical observables-can be characterized and controlled by means of a large-deviation formalism.