For certain Veech surfaces, the paper shows that the Teichmüller geodesic flow pushes twist tori to dense sets, and under an algebraic monodromy condition every weak-* limit is fully supported.
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Veech Surfaces and Expanding Twist Tori on Moduli Spaces of Abelian Differentials
For certain Veech surfaces, the paper shows that the Teichmüller geodesic flow pushes twist tori to dense sets, and under an algebraic monodromy condition every weak-* limit is fully supported.