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.
Monotonicity of maximal equicontinuous factors and an application to toral flows
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abstract
We show that for group actions on locally connected spaces the maximal equicontinuous factor map is always monotone, that is, the preimages of single points are connected. As an application, we obtain that if the maximal continuous factor of a homeomorphism of the two-torus is minimal, then it is either (i) an irrational translation of the two-torus, (ii) an irrational rotation on the circle or (iii) the identity on a singleton.
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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.