In this talk, we show a recent progress about Hausdorff dimension of the set of nonergodic directions. Let X be the resulting surface by gluing two copies of the flat torus along a segment with holonomy vector (lambda,mu) and let q_k be the sequence of best simultaneous approximation denominators to (lambda,mu)，related to any norm of R^2. If q_{k+1}=O(q_k^N) for some N>0, then the set of nonergodic directions in X has Hausdorff dimension 1/2; if /sum(loglogq_{k+1})/q_k=infty, then the dimension is 0. This was a joint work with Yitwah Cheung.

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