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The unique ergodic measure of the symmetric piecewise toral isometry of rotation angle θ=kπ/5 is the Hausdorff measure of its singular set

It is known that the symmetric piecewise toral isometry of rotation angle θ = k π /5, k=1, 2, 3, 4 is uniquely ergodic in ‘a certain subset’ of its singular set (aka exceptional set). The purpose of this paper is to identify the unique ergodic measure explicitly. In fact, we prove that the unique ergodic measure is none other than the normalized Hausdorff measure of the singular set, consequently proving that the unique ergodicity holds in the entire singular set. We use the ‘phantom dynamics’ given by a number of symmetry identifications as our main tool.