The distance in Riemann manifold

Question

Let $f: M\to M$, where $(M,\rho)$ is a closed Riemann manifold, and $(\widetilde{M},\widetilde{\rho})$ is the universal covering of $(M,\rho)$, $D$ is a fundamental domain of $(\widetilde{M},\widetilde{\rho})$. $\widetilde{f}$ is the lift of $f$, we suppose there is $x\in D$ s.t. $\widetilde{f}(x)=x$ then I want to prove that: there exist a constant $C$, for all $n\in \mathbb{N}$, $$max\{max_{x\in M}|d_xf^n|,max_{x\in M}|d_xf^{-n}|\} \ge C sup_{z\in D}distance_{\widetilde{\rho}}(x,\widetilde{f}^nz).$$

Answer 1

Yes, $C=1$ and you do not even need to include derivatives of inverse iterations in the left hand side. To prove this just use the definition of the length of the image under f of a geodesic in M connecting projection of x to projection of z.