  We study the evolution of distributions under the action of an ergodic
dynamical system, which may be stochastic in nature. By employing tools from
Koopman and transfer operator theory one can evolve any initial distribution of
the state forward in time, and we investigate how estimators of these operators
perform on long-term forecasting. Motivated by the observation that standard
estimators may fail at this task, we introduce a learning paradigm that neatly
combines classical techniques of eigenvalue deflation from operator theory and
feature centering from statistics. This paradigm applies to any operator
estimator based on empirical risk minimization, making them satisfy learning
bounds which hold uniformly on the entire trajectory of future distributions,
and abide to the conservation of mass for each of the forecasted distributions.
Numerical experiments illustrates the advantages of our approach in practice.
