A piecewise linear circle homeomorphism with rational rotation number is conjugate to a rigid rational rotation exactly when every break point is periodic, and near such parameter values in families the rotation number scales linearly.
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Piecewise linear circle maps and conjugation to rigid rational rotations
A piecewise linear circle homeomorphism with rational rotation number is conjugate to a rigid rational rotation exactly when every break point is periodic, and near such parameter values in families the rotation number scales linearly.