Anosov closing lemma: how to control the exact period

Question

Question: I would like to state and prove Anosov's closing lemma in such a way that we know the exact period of the orbit.

Framework: $\varphi : \mathbb R^n \longrightarrow \mathbb R^n$ is a smooth diffeomorphism and $\Lambda \subset \mathbb R^n$ is a hyperbolic set. Here is how I was presented Anosov's closing lemma :

Take $\epsilon, \delta > 0$ as in the shadowing lemma. Suppose that for some $x \in \Lambda$ there exists an $N \in \mathbb N^*$ such that $$ || \varphi^N(x) - x|| < \delta $$ Then there exists a periodic orbit of $\varphi$ that is in $\Lambda + B(0,\epsilon)$.

Here is how I state Anosov's shadowing lemma:

There exists $\epsilon_0 > 0$ such that forall $\epsilon \in (0, \epsilon_0)$, there exists $\delta > 0$ such that forall $(q_j)_{j \in \mathbb Z}$ $\delta$-pesudo orbit of $\varphi$ that is in $\Lambda + B(0,\delta)$, there exists a unique $\epsilon$-shadowing orbit of $(q_j)$.

The idea of the proof of the closing lemma is to consider the $\delta$-pseudo orbit $q_j = \varphi^{j ~\mathrm{mod} ~N}(x)$ and apply the shadowing lemma. By uniqueness one can prove the resulting shadowing orbit is $N$-periodic.

Discussion: In the last idea of proof one don't prove that $N$ is the exact period. Considering $x$ fixed point of $\varphi$ and $N > 1$ it is clear we cannot ask for $N$ to be the period. I think I can achieve my goal asking $N$ to be minimal in $\mathbb N^*$. I don't see how to complete the proof of this last claim, in my opinion I don't have enough information about how the periodic point behaves as a function of the pseudo-orbit. In my course we have not covered the proof of the shadowing lemma. Any hint or reference to a proof including ideas on how to control the exact period are welcome.

Answer 1

The standard reference is the book of Katok-Hasselblatt, introduction to the modern theory of dynamical systems. The closing lemma is Theorem 6.4.15:

Let $A$ be a hyperbolic set for $f:U \rightarrow M$. Then there exists an open neighborhood $V$ containing $A$ and $C$, $\varepsilon > 0$ such that for $\varepsilon < \varepsilon_0$ and any periodic $\varepsilon$-orbit $(x_0,..., x_m) \in V$, there is a point $y\in U$ such that $f^m(y) = y$ and $dist(f^k(y), x_k) < C\varepsilon$ for $k = 0,..., m-1$.

If $\epsilon$ is small with respect to the mutual distances $d(x_i, x_j)$, the $x_j$ being assumed to be mutually distinct, then since the point $f^i(y)$ is very close to $x_i$, it cannot be close to $x_j$ and so neither to $f^j(y)$. This guarantees that the smallest period of $y$ is $m$. This is better seen on a figure, with small disjoint balls around the $x_i's$ and the $f^i(y)$ belonging to these balls.