A non-annular torus homeomorphism with small wandering domains admits an irrational circle factor exactly when it has uniformly bounded rotational deviations in some rational direction.
Addas-Zanata, On properties of the vertical rotation interval for twist mappings, Ergodic Theory and Dynamical Systems 25 (2005), no
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Periodic point free homeomorphisms and irrational rotation factors
A non-annular torus homeomorphism with small wandering domains admits an irrational circle factor exactly when it has uniformly bounded rotational deviations in some rational direction.