Exercise: Let $\;p\in\mathcal{U}\subset\mathbb{R}^{d}$ a periodic point and $P:S'\rightarrow S$ the Poincaré transformation associated to transversal section $S$ in $p$. Let $z\in S$ such that $P^n(z)=f^{t_n}(z)$ is defined for all $n\geq0$ and \begin{equation} \|P^n(z)-p\|\leq C\lambda^{n}, \;\;\;\;\;\;\;\;\;\;\;(1) \end{equation} with $C>0$ and $\lambda<1$. Show that the orbit of $z$ is assymptotic to orbit of $f^r(p)$, for some $r\in\mathbb{R}$.
Attempt:
Given $P^n(z)=f^{t_n}(z)$, we have $\tau(P^n(z))=t_{n+1}-t_n$, where $\tau:S'\rightarrow\mathbb{R}$ is the return time function, and $\tau\in C^k$.$\;$ As $\tau\in C^k$, then $\;t_{n+1}-t_n \rightarrow \tau(p)=T$, where $T>0\;$ is the period. In particular, exists $n_0\geq1$ such that, for all $n\geq n_0$ $$t_{n+1}-t_n < T+1.$$ By Continuous Dependence Theorem, given $\epsilon>0$, exists $\delta>0$ such that $$\|x-p\| \leq\delta \Rightarrow \|f^s(x)-f^s(p)\|\leq\epsilon,\; \forall s\in[0,T+1].$$ The condition $(1)$ says that $P^n(z)$ converges exponentially to $p$, but I don't know how use this to prove.
Any help would be appreciated!