Abstract
Let X be a genus two double covering surface of the standard flat torus $$({\mathbb {C}}/{\mathbb {Z}}[i],dz)$$ with two ramified values $$0\ mod\ {\mathbb {Z}}[i]$$ and $$\lambda +i\mu \ mod\ {\mathbb {Z}}[i]$$ . We prove that if the equation $$l+m\lambda +n\mu =0$$ has nonzero integer solutions then the Hausdorff dimension of the set of nonergodic directions of X is either 0 or 1/2. And the precise criterion for the two possibilities is provided.
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