Dynamics of kaleidoscopic maps |
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Authors: | Byungik Kahng |
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Institution: | Department of Mathematics, The College of William and Mary, 114 Jones Hall P.O. Box 8795, Williamsburg, VA 23187-8795, USA |
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Abstract: | First, we introduce a certain class of piecewise affine elliptic rotation maps on , called the kaleidoscopic maps, and describe its importance.And then, we concentrate our efforts on a special case, when the rotation angle θ of a kaleidoscopic map is , . For the special case, we answer the conjectures regarding the periodicity and the singularity structure of such (kaleidoscopic) dynamics. In the process, we prove the partial riddling of the regular orbits that gives rise to the classification of periodic sets, and estimate the Hausdorff dimension of the singular set.Finally, we study the dynamics of such kaleidoscopic maps restricted within the singular set, and answer conjectures concerning the chaos, the local chaos, and the ergodicity with respect to the normalized Hausdorff measure of the singular set. |
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Keywords: | Primary: 37E99 28A80 Secondary: 37A25 37A35 37J10 37J15 37J20 37J25 65P10 65P20 65P40 65P30 94A12 |
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