American Institute of Mathematical Sciences

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February  2005, 13(2): 515-532. doi: 10.3934/dcds.2005.13.515

Double rotations

Hideyuki Suzuki 1, , Shunji Ito 2, and  Kazuyuki Aihara 3,

1. 

Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
2. 

Department of Information and Systems Engineering, Faculty of Engineering, Kanazawa University, 2-40-20 Kodatsuno, Kanazawa 920-8667, Japan
3. 

Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan

Received  March 2003 Revised  April 2005 Published  April 2005

We consider a map called a double rotation, which is composed of two rotations on a circle. Specifically, a double rotation is a map on the interval $[0,1)$ that maps $x\in[0,c)$ to $\{x+\alpha\}$, and $x\in[c,1)$ to $\{x+\beta\}$. Although double rotations are discontinuous and noninvertible in general, we show that almost every double rotation can be reduced to a simple rotation, and the set of the parameter values such that the double rotation is irreducible to a rotation has a fractal structure. We also examine a characteristic number of a double rotation, which is called a discharge number. The graph of the discharge number as a function of $c$ reflects the fractal structure, and is very complicated.
Keywords: rotation, piecewise isometry., Double rotation, reduction theory, interval translation mapping.
Mathematics Subject Classification: 37E05, 37E10, 37B0.
Citation: Hideyuki Suzuki, Shunji Ito, Kazuyuki Aihara. Double rotations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 515-532. doi: 10.3934/dcds.2005.13.515
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