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

Abstract

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.

Acknowledgments

This paper was written during the author's appointment in the University of Minnesota at Morris. The author thanks (in alphabetical order) Roy Adler, Peter Ashwin, Keith Burns, William Galway, Arek Goetz, John Lowenstein, Dan Mauldin, and Julian Palmore for their valuable comments. The author also appreciates the referees' suggestions to include the computation of the Hausdorff measures, which is presented in the appendix, and to include a more detailed explanation on the graph-directed iteration function system, through which the computation of the Hausdorff dimension and the normalization of the Hausdorff measure are carried out.

Notes

The additional identifications with respect to some periodic points are the essence of the ‘phantom dynamics’. It is an extension of the midpoint symmetry identification method in Kahng (2000, 2002) and Adler et al. (2001) that helped characterize the singularity structure of the θ=π/4 case.

Especially, it must be noted that this step is not related to Theorem 7 that allows us to normalize the Hausdorff measure.