Let $f\colon \mathbb{R}^{2}\to\mathbb{R}^{2}$ be a $C^{1}$-diffeomorphism. Let $\|\cdot\|$ denote the operator norm. Let $p$ and $q$ be such that $$\lim_{n\to\infty}f^{-n}(q)=\lim_{n\to\infty}f^{n}(q)=p.$$ Also suppose that there exists a $0<\lambda<1$ such that $$\|D(f^{-1})(p)\|<\lambda,\qquad\|Df(p)\|<\lambda.$$
Does there exist a $C>0$ such that $\|D(f^{-n})(f^{k}(q))\|\leq C\lambda^{n}$ and $\|D(f^{n})(f^{k}(q))\|\leq C\lambda^{n}$ for all $k\in\mathbb{Z}$ and $n\in\mathbb{N}$?
I have shown that there exists an integer $N\in\mathbb{N}$ such that $$\|D(f^{-1})(f^{k}(q))\|<\lambda,\qquad\|Df(f^{k}(q))\|<\lambda$$ for all $k\in\mathbb{Z}$ with $|k|\geq N$. Also, by the chainrule we have $$\|D(f^{-n})(f^{k}(q))\|\leq\prod_{i=1}^{n}\|D(f^{-1})(f^{k-i}(q))\|$$ and a similar estimation for $\|D(f^{n})(f^{k}(q))\|$.
Is my reasoning right? And how do I finish the proof?