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Double rotations

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  • 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.
    Mathematics Subject Classification: 37E05, 37E10, 37B05.


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