I have to prove that the diffeomorphism $f(x) = x^3+x$ is topologically conjugated to a linear map. Thanks to the study of its orbits, I know that the linear map $l : x \to ax$ must satisfy $a > 1$ and that any such map will work, since they are topologically conjugated.
However, I wasn't able to explicit any homeorphism $\varphi : \mathbb{R} \to \mathbb{R}$ such that $\varphi \circ f = l \circ \varphi$ and I can't find any theorem of existence since $f$ isn't topologically linearizable at $0$. From next question, I know that $\varphi$ can't be bi-Hölder, if that can be of any help in the search of an explicit function.
Do you have any hint ?
