Non-uniqueness of composition square roots (Q1852423)

Let \(g\) be defined on a neighborhood of a fixed point \(0\). A composition square root of \(g\) is a function \(f\) satisfying \(f^2=g\) on some neighborhood \(I\) of the fixed point \(0\). It is shown that for each \(\mu >0\) there is a differentiable and nonlinearizable interval map \(g\) with nonvanishing derivative defined on a neighborhood of a fixed point \(0\) with \(g'(0)=\mu\) such that \(g\) has infinitely many differentiable and nonlinearizable orientation-reversing composition square roots with nonvanishing first derivatives on a neighborhood of \(0\).