  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
work, 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 work 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.
