What about the differentiability of two conjugate mappings?

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Let $M$ be a smooth manifold, and $f,g:M\rightarrow M$ be topologically conjugate(i.e., there exists a homeomorphism $h:M\rightarrow M$ such that $f\circ h=h\circ g$). If $f,g$ are differentiable, is it possible that $f,g$ have different order of differentiability?