Geodesic
A general class of iterative equations on the unit circle
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 3, pp. 809-829.

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A class of functional equations with nonlinear iterates is discussed on the unit circle ${\mathbb{T}}^1$. By lifting maps on ${\mathbb{T}}^1$ and maps on the torus ${\mathbb{T}}^n$ to Euclidean spaces and extending their restrictions to a compact interval or cube, we prove existence, uniqueness and stability for their continuous solutions.
Classification : 37E05, 39B12, 39B22, 39B32, 39B82
Keywords: iterative equation; circle; lift; orientation-preserving; continuation 
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     author = {Zdun, Marek C. and Zhang, Weinian},
     title = {A general class of iterative equations on the unit circle},
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Zdun, Marek C.; Zhang, Weinian. A general class of iterative equations on the unit circle. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 3, pp. 809-829. http://geodesic.mathdoc.fr/item/CMJ_2007__57_3_a2/