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.
The Franks-Misiurewicz conjecture for extensions of irrational rotations
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abstract
We show that a toral homeomorphism which is homotopic to the identity and topologically semiconjugate to an irrational rotation of the circle is always a pseudo-rotation (i.e. its rotation set is a single point). In combination with recent results, this allows us to complete the study of the Franks-Misiurewicz conjecture in the minimal case.
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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.