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We show that for group actions on locally connected spaces the maximalequicontinuous factor map is always monotone, that is, the preimages of singlepoints are connected. As an application, we obtain that if the maximalcontinuous factor of a homeomorphism of the two-torus is minimal, then it iseither (i) an irrational translation of the two-torus, (ii) an irrationalrotation on the circle or (iii) the identity on a singleton.

Monotonicity of maximal equicontinuous factors and an application to toral flows