I'm currently working on a question on chaotic dynamics on fractals. Particularly if we have an iterated function system of conformal differentiable contraction mappings {$f_1$,...,$f_n$} we denote the contraction ratio of the function $f_i$ by $||f_i'||_\infty$ and similarly if $\omega=\omega_1\omega_2...\omega_k$ is a finite word over the alphabet {1,...,n} then $f_\omega=f_{\omega_1}\circ f_{\omega_2}\circ \cdots\circ f_{\omega_k}$ and the contraction ratio is $||f'_\omega||_\infty\leq||f'_{\omega_1}||_\infty\cdots||f'_{\omega_k}||_\infty$ by submultiplicity.
Define $\omega^-$ to be the word $\omega_1\omega_2...\omega_{k-1}$ and $r_{min}=\min_i ||f'_{i}||_\infty\in(0,1)$.
My question is do we have $||f'_{\omega^-}||_\infty r_{min}\leq ||f'_\omega||_\infty$?
I'm happy to give more context to this problem if that will help anyone answering.