A continuously differentiable map $f:S^{1}\rightarrow S^{1}$ is called expanding if $|f^{'}(x)|>1$ for all $x\in S^{1}$.

We can define the degree of f, def(f) to be number of preimage $f^{-1}(x)$, for any $x\in S^{1}.$

**Theorem** Any expanding map $f:S^{1}\rightarrow S^{1}$ of degree $d\geq2$ is topologically conjugate to linear of degree d.

**Question**Let f be $C^{1+\epsilon}$. Does the theorem work for $C^{1+\epsilon}$ conjugacy?if No,Does anyone know under what assumption we have $C^{1+\epsilon} $conjugacy above theorem?

**Remark** We have following theorem:
Let $2\leq r\leq t$. If two orientation preserving expanding $C^r$ endomorphisms
f and g of $S^{l}$ are absolutely continuously conjugate, then they are conjugate by a $C^r$
diffeomorphism. But, i do not assume $C^{2}$. In fact, i was asking. Does Shub and Sullivan's theorem work for $C^{1+\epsilon}?