## Appendix A Proofs of Theorems and Corollaries  
Proof of Theorem 1.: Each of the proof for the case \(k=N\), or \(T\), is given as follows. 1. In the case of \(k=N\): By using Equations (5) and (8), the break points \((-(1-\xi)b_{p},f_{N}(-(1-\xi)b_{p},\zeta,\eta,\zeta))\) and \(((\eta+\xi)b_{p},f_{N}((\eta+\xi)b_{p},\zeta,\eta,\zeta))\) are transformed into \((\psi_{N1},\phi_{N1})\) and \((\psi_{N2},\phi_{N2})\), respectively, such that the following is the case. \[\psi_{N1} = \frac{1}{2}\biggl{(}-(1-\xi)+\frac{\zeta}{r}\biggr{)}b_{p},\ \phi_{N1}=\frac{1}{2}\biggl{(}(1-\xi)+\frac{\zeta}{r}\biggr{)}b_{p},\] \[\psi_{N2} = \frac{1}{2}\biggl{(}\eta+\xi-\frac{\zeta\eta}{r}\biggr{)}b_{p},\ \phi_{N2}=\frac{1}{2}\biggl{(}-(\eta+\xi)-\frac{\zeta\eta}{r}\biggr{)}b_{p}.\]  
\(\psi_{N2}>\psi_{N1}\) and \(\phi_{N1}>\phi_{N2}\) hold because \(\psi_{N2}-\psi_{N1}=\frac{b_{p}(\eta+1)(r-\zeta)}{2r}>0\) and \(\phi_{N1}-\phi_{N2}=\frac{b_{p}(\eta+1)(r+\zeta)}{2r}>0\), respectively. Since \((\psi,\phi)\)-coordinate system is identical with the \(-\pi/4\) rotation of \((v,i)\)-coordinate system and \(\psi_{N2}>\psi_{N1}\) and \(\phi_{N1}>\phi_{N2}\), as shown in Figure A1, \(\phi(\tau)\) is explicitly described by a three-segment piecewise linear function of Equation (16). \(0<1-r<1\), \(1<1+r<2\), \(0<r-\zeta<r\), and \(r+\zeta<2r\) hold because of the conditions such that \(1>r>\zeta>0\) and \(\eta>1\). Therefore, \(0<D\chi_{N1}=\frac{(1-r)}{(r+1)}<1\), \(-1>D\chi_{N2}=\frac{(\zeta+r)}{(\zeta-r)}\), and \(0<D\chi_{N3}=\frac{(1-r)}{(r+1)}<1\) hold. In the case of \(k=T\): By using Equations (6) and (8), the break points \(((1+\xi)b_{p},f_{T}((1+\xi)b_{p},\zeta,\eta,\xi))\) and \(((\eta+\xi)b_{p},f_{T}((\eta+\xi)b_{p},\zeta,\eta,\xi))\) are transformed into \((\psi_{T1},\phi_{T1})\) and \((\psi_{T2},\phi_{T2})\), respectively, such that the following is the case:  
\[\psi_{T1} = \frac{1}{2}\Big{(}1+\xi+\frac{1}{r}\Big{)}b_{p},\ \phi_{T1}=\frac{1}{2}\Big{(}-1-\xi+\frac{1}{r}\Big{)}b_{p},\] \[\psi_{T2} = \frac{1}{2}\Big{(}\eta+\xi+\frac{1-\xi(\eta-1)}{r}\Big{)}b_{p},\] \[\phi_{T2} = \frac{1}{2}\Big{(}-\eta-\xi+\frac{1-\xi(\eta-1)}{r}\Big{)}b_{p}.\]  
\(\psi_{T2}>\psi_{T1}\) and \(\phi_{T1}>\phi_{T2}\) hold because \(\psi_{T2}-\psi_{T1}=\frac{b_{p}(\eta-1)(r-\xi)}{2r}>0\) and \(\phi_{T1}-\phi_{T2}=\frac{b_{p}(\eta-1)(r+\xi)}{2r}>0\), respectively. Since the \((\psi,\phi)\)-coordinate system is identical with the \(-\pi/4\) rotation of \((v,i)\)-coordinate system and \(\psi_{T2}>\psi_{T1}\) and \(\phi_{T1}>\phi_{T2}\), as shown in Figure A2, \(\phi(\tau)\) is explicitly described by a three-segment piecewise linear function of Equation (16). \(0<1-r<1,1<1+r<2,0<r-\zeta<r\), \(r+\xi<2r\), \(1<2-r<2\), and \(2<r+2<3\) hold because of the conditions such that \(1>r>\zeta>0\) and \(\eta>1\). Therefore, \(0<D\chi_{T1}=\frac{(1-r)}{(r+1)}<1\), \(-1>D\chi_{T2}=\frac{(\zeta+r)}{(\xi-r)}\), and \(0<D\chi_{T3}=\frac{(2-r)}{(r+2)}<1\) hold. The proof is now complete. Proof of Corollary 1.: With the substitution of Equations (16) into (9), \(\psi(\tau)=h_{k}(\psi(\tau-1))\), that is, Equation (17) is easily obtained. Then, \(0<1-R<1\) and \(1<1+R<2\), that is, \(0<\frac{1-R}{1+R}<1\) hold because \(1>R>0\). Therefore, under the conditions of Corollary 1, \(0<Dh_{k1}=\frac{(1-R)D\chi_{k1}}{1+R}<1\), \(-1>Dh_{k2}=\frac{(1-R)D\chi_{k2}}{1+R}\), and \(0<Dh_{k3}=\frac{(1-R)D\chi_{k3}}{1+R}<1\) hold. Proof of Corollary 2.: The 1-D map in Corollary 2 is easily derived from Equations (7), (8), and (10); Corollary 1; and Theorem 1. **Proof of Theorem 2. ** Under the conditions that Theorem 2 holds, the following is the case. In the case of \(k=T\): It is easy to check the following properties: (1) \(\psi_{T*}>0\) holds because \(r-\zeta>0\) and \(r-R\zeta>0\).