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.

中文翻译：

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非遍历方向集的Hausdorff维数二分法

令X为标准平坦圆环\（（{{mathbb {C}} / {\ mathbb {Z}} [i]，dz）\）的两个双覆盖面，具有两个分母值\（0 \ mod \ {\ mathbb {Z}} [i] \）和\（\ lambda + i \ mu \ mod \ {\ mathbb {Z}} [i] \）。我们证明，如果方程\（l + m \ lambda + n \ mu = 0 \）具有非零整数解，则X的非遍历方向集的Hausdorff维数为0或1/2。并提供了两种可能性的精确标准。