[proofplan]
We work in a chart centred at the fixed point and identify the tangent space with the hyperbolic splitting $E^s \oplus E^u$. In these coordinates the map is its block diagonal linear part plus a nonlinear remainder whose $C^1$ norm is made small by shrinking the chart. A standard Hadamard graph transform theorem then produces a unique invariant $C^r$ graph over the stable subspace, identifies it with the points whose forward orbits remain in a small product neighbourhood, and gives exponential contraction. Applying the same construction to $f^{-1}$ gives the unstable disk, and local equivalence between a Riemannian distance and the coordinate norm transfers the estimates back to $M$.
[/proofplan]
[step:Choose coordinates adapted to the hyperbolic splitting]
Let
\begin{align*}
A := df_p: T_pM \to T_pM
\end{align*}
be the derivative of $f$ at $p$. Since $E^s$ and $E^u$ are $A$-invariant and $T_pM=E^s \oplus E^u$, write
\begin{align*}
A_s := A|_{E^s}: E^s \to E^s, \qquad A_u := A|_{E^u}: E^u \to E^u.
\end{align*}
The hyperbolicity hypothesis implies that every eigenvalue of $A_s$ has modulus less than $1$, and every eigenvalue of $A_u^{-1}$ has modulus less than $1$.
Choose norms $|\cdot|_s$ on $E^s$ and $|\cdot|_u$ on $E^u$, and choose a number $\lambda \in (0,1)$, such that
\begin{align*}
|A_s \xi|_s \le \lambda |\xi|_s \quad \text{for every } \xi \in E^s
\end{align*}
and
\begin{align*}
|A_u^{-1}\eta|_u \le \lambda |\eta|_u \quad \text{for every } \eta \in E^u.
\end{align*}
Equivalently,
\begin{align*}
|A_u\eta|_u \ge \lambda^{-1}|\eta|_u \quad \text{for every } \eta \in E^u.
\end{align*}
On $E := E^s \oplus E^u$, define the product norm
\begin{align*}
|(\xi,\eta)| := \max\{|\xi|_s,|\eta|_u\}.
\end{align*}
Choose a $C^r$ chart
\begin{align*}
\varphi: V \to \varphi(V) \subset E
\end{align*}
with $p \in V$, $\varphi(p)=0$, and $d\varphi_p: T_pM \to E$ equal to the identity under the decomposition $T_pM=E^s \oplus E^u$. Shrinking $V$ if necessary, define the coordinate representative
\begin{align*}
F: \varphi(V \cap f^{-1}(V)) \to E, \qquad F(z) := \varphi(f(\varphi^{-1}(z))).
\end{align*}
Then $F(0)=0$ and $DF_0=A$. Hence, for $z=(\xi,\eta) \in E^s \oplus E^u$ near $0$,
\begin{align*}
F(\xi,\eta) = (A_s\xi + R_s(\xi,\eta), A_u\eta + R_u(\xi,\eta)),
\end{align*}
where
\begin{align*}
R: \varphi(V \cap f^{-1}(V)) \to E, \qquad R(z):=F(z)-Az,
\end{align*}
and $R_s: \varphi(V \cap f^{-1}(V)) \to E^s$, $R_u: \varphi(V \cap f^{-1}(V)) \to E^u$ are the components of $R$ in the splitting $E=E^s \oplus E^u$. Since $DR_0=0$, after shrinking $V$ again we may assume that $R$ has arbitrarily small $C^1$ norm on a product ball around $0$.
[/step]
[step:Construct the stable disk by the Hadamard graph transform]
Choose $\varepsilon>0$ sufficiently small, depending only on $\lambda$, and shrink $V$ so that
\begin{align*}
\|DR_z\|_{\mathcal{L}(E,E)} \le \varepsilon
\end{align*}
for every $z$ in a product ball
\begin{align*}
Q_\rho := \{(\xi,\eta) \in E^s \oplus E^u : |\xi|_s < \rho,\ |\eta|_u < \rho\}
\end{align*}
with $\rho>0$. We use the standard Hadamard graph transform theorem in the following form: if a $C^r$ map $F$ on $Q_\rho$ has the above hyperbolic block form, with $A_s$ contracting, $A_u^{-1}$ contracting, and $\|DR\|_{C^0}$ sufficiently small, then there exists a unique $C^r$ map
\begin{align*}
h_s: B_s(\rho) \to E^u
\end{align*}
defined on
\begin{align*}
B_s(\rho) := \{\xi \in E^s : |\xi|_s < \rho\}
\end{align*}
such that $h_s(0)=0$, $Dh_s(0)=0$, $|h_s(\xi)|_u<\rho$, and
\begin{align*}
\Gamma_s := \{(\xi,h_s(\xi)) : \xi \in B_s(\rho)\}
\end{align*}
is locally forward invariant and satisfies the maximality property
\begin{align*}
\Gamma_s = \{z \in Q_\rho : F^n(z) \in Q_\rho \text{ for every } n \in \mathbb{N}_0\}.
\end{align*}
The same theorem gives constants $K_s \ge 1$ and $\theta_s \in (0,1)$ such that
\begin{align*}
|F^n(z)| \le K_s\theta_s^n |z|
\end{align*}
for every $z \in \Gamma_s$ and every $n \in \mathbb{N}_0$.
Here the quoted graph transform theorem is the standard Hadamard graph transform theorem for a hyperbolic fixed point in finite dimensions (citing a result not yet in the wiki: Hadamard graph transform theorem). Its proof applies the [contraction mapping theorem](/theorems/71) to the transform sending a Lipschitz graph over $E^s$ to the stable-coordinate projection of its forward image; the domination between $A_s$ and $A_u$ makes this transform a contraction. The $C^r$ regularity follows because the graph transform preserves $C^r$ graphs and the fixed graph is obtained as the limit in the corresponding $C^r$ graph space. If $r=\infty$, applying the finite-$m$ statement for each $m \in \mathbb{N}$ gives that the same fixed graph is $C^m$ for every $m$, hence $C^\infty$.
[guided]
The point of the coordinate preparation is to put the fixed point into the standard form required by the graph transform. We have a product decomposition $E=E^s \oplus E^u$, and in this decomposition the coordinate map has the form
\begin{align*}
F(\xi,\eta) = (A_s\xi + R_s(\xi,\eta), A_u\eta + R_u(\xi,\eta)).
\end{align*}
The inequalities
\begin{align*}
|A_s\xi|_s \le \lambda |\xi|_s
\end{align*}
and
\begin{align*}
|A_u^{-1}\eta|_u \le \lambda |\eta|_u
\end{align*}
say that the stable linear part contracts forward and the unstable linear part contracts backward. Equivalently, the unstable linear part expands forward:
\begin{align*}
|A_u\eta|_u \ge \lambda^{-1}|\eta|_u.
\end{align*}
The nonlinear term $R$ satisfies $R(0)=0$ and $DR_0=0$. Since $DR$ is continuous, shrinking the coordinate neighbourhood makes
\begin{align*}
\|DR_z\|_{\mathcal{L}(E,E)} \le \varepsilon
\end{align*}
on the product ball $Q_\rho$, where $\varepsilon>0$ is chosen small compared with the spectral gap between $\lambda$ and $\lambda^{-1}$. This is the hypothesis that says the nonlinear map is a small perturbation of the hyperbolic [linear map](/page/Linear%20Map).
Now apply the Hadamard graph transform theorem in finite dimensions. The theorem considers graphs over the stable ball,
\begin{align*}
\Gamma_g := \{(\xi,g(\xi)) : \xi \in B_s(\rho)\},
\end{align*}
where $g: B_s(\rho)\to E^u$ is Lipschitz with small Lipschitz constant. The image $F(\Gamma_g)$ can again be written locally as a graph over $E^s$ because the stable coordinate map is a small perturbation of $\xi \mapsto A_s\xi$, while the unstable expansion prevents two distinct unstable fibres from producing the same bounded forward orbit. This defines the graph transform.
The domination between the estimates for $A_s$ and $A_u$ makes the graph transform a contraction on a complete space of Lipschitz graphs. Therefore it has a unique fixed graph. Denote its graphing map by
\begin{align*}
h_s: B_s(\rho) \to E^u.
\end{align*}
The fixed graph is
\begin{align*}
\Gamma_s := \{(\xi,h_s(\xi)) : \xi \in B_s(\rho)\}.
\end{align*}
The fixed point property of the graph transform gives local forward invariance of $\Gamma_s$.
The graph transform theorem also identifies the fixed graph with the maximal set of points whose forward iterates remain in the product neighbourhood:
\begin{align*}
\Gamma_s = \{z \in Q_\rho : F^n(z) \in Q_\rho \text{ for every } n \in \mathbb{N}_0\}.
\end{align*}
This is the isolating-neighbourhood part of the argument: the equality is asserted only after choosing $Q_\rho$ sufficiently small and product-shaped in the stable-unstable splitting.
Finally, the same theorem gives exponential contraction along the fixed graph. Thus there exist constants $K_s \ge 1$ and $\theta_s \in (0,1)$ such that
\begin{align*}
|F^n(z)| \le K_s\theta_s^n |z|
\end{align*}
for every $z \in \Gamma_s$ and every $n \in \mathbb{N}_0$. The graph is $C^r$ because the transform acts on $C^r$ graphs and its fixed point lies in that class. When $r=\infty$, the finite-$m$ conclusion for every $m$ gives smoothness.
[/guided]
[/step]
[step:Transfer the stable graph back to the manifold]
Define
\begin{align*}
U_s := \varphi^{-1}(Q_\rho)
\end{align*}
and
\begin{align*}
W^s_{\mathrm{loc}}(p) := \varphi^{-1}(\Gamma_s).
\end{align*}
Since $h_s: B_s(\rho)\to E^u$ is $C^r$, the map
\begin{align*}
\iota_s: B_s(\rho) \to M, \qquad \iota_s(\xi):=\varphi^{-1}(\xi,h_s(\xi))
\end{align*}
is a $C^r$ embedding onto $W^s_{\mathrm{loc}}(p)$. Hence $W^s_{\mathrm{loc}}(p)$ is an embedded $C^r$ disk of dimension $\dim E^s$.
Because $h_s(0)=0$, one has $p=\varphi^{-1}(0,0) \in W^s_{\mathrm{loc}}(p)$. Since $Dh_s(0)=0$, the tangent space of the graph at $0$ is
\begin{align*}
T_0\Gamma_s = E^s \oplus \{0\}.
\end{align*}
Under the identification induced by $d\varphi_p=\operatorname{id}_{T_pM}$, this gives
\begin{align*}
T_pW^s_{\mathrm{loc}}(p)=E^s.
\end{align*}
The local forward invariance of $\Gamma_s$ gives, after replacing $U_s$ and $W^s_{\mathrm{loc}}(p)$ by smaller neighbourhoods of $p$ if needed,
\begin{align*}
f(W^s_{\mathrm{loc}}(p)) \subset W^s_{\mathrm{loc}}(p).
\end{align*}
Moreover, the maximality property of $\Gamma_s$ gives
\begin{align*}
W^s_{\mathrm{loc}}(p)=\{x \in U_s : f^n(x)\in U_s \text{ for every } n \in \mathbb{N}_0\}.
\end{align*}
[/step]
[step:Construct the unstable disk by applying the stable construction to $f^{-1}$]
Apply the preceding stable construction to the $C^r$ diffeomorphism
\begin{align*}
f^{-1}: M \to M.
\end{align*}
The derivative of $f^{-1}$ at $p$ is
\begin{align*}
d(f^{-1})_p = A^{-1}.
\end{align*}
For $A^{-1}$, the stable generalized eigenspace is $E^u$, because the eigenvalues of $A$ on $E^u$ have modulus greater than $1$, and the unstable generalized eigenspace is $E^s$.
Thus there exist $\rho_u>0$, a product neighbourhood
\begin{align*}
Q_{\rho_u}^{u} := \{(\xi,\eta)\in E^s\oplus E^u : |\xi|_s<\rho_u,\ |\eta|_u<\rho_u\},
\end{align*}
and a $C^r$ graph over $E^u$,
\begin{align*}
\Gamma_u := \{(h_u(\eta),\eta): \eta \in B_u(\rho_u)\},
\end{align*}
where
\begin{align*}
B_u(\rho_u):=\{\eta \in E^u: |\eta|_u<\rho_u\}
\end{align*}
and
\begin{align*}
h_u: B_u(\rho_u)\to E^s
\end{align*}
satisfies $h_u(0)=0$ and $Dh_u(0)=0$. Define
\begin{align*}
U_u := \varphi^{-1}(Q_{\rho_u}^{u})
\end{align*}
and
\begin{align*}
W^u_{\mathrm{loc}}(p):=\varphi^{-1}(\Gamma_u).
\end{align*}
Then $W^u_{\mathrm{loc}}(p)$ is an embedded $C^r$ disk of dimension $\dim E^u$, contains $p$, and satisfies
\begin{align*}
T_pW^u_{\mathrm{loc}}(p)=E^u.
\end{align*}
Its local invariance for $f^{-1}$ is
\begin{align*}
f^{-1}(W^u_{\mathrm{loc}}(p)) \subset W^u_{\mathrm{loc}}(p),
\end{align*}
after shrinking if necessary. Its maximality property is
\begin{align*}
W^u_{\mathrm{loc}}(p)=\{x \in U_u : f^{-n}(x)\in U_u \text{ for every } n \in \mathbb{N}_0\}.
\end{align*}
The graph transform estimate for $f^{-1}$ also gives constants $K_u\ge 1$ and $\theta_u\in(0,1)$ such that
\begin{align*}
|F^{-n}(z)| \le K_u\theta_u^n |z|
\end{align*}
for every $z\in \Gamma_u$ and every $n\in\mathbb{N}_0$.
[/step]
[step:Choose one neighbourhood and convert the coordinate estimates to Riemannian distance estimates]
Let
\begin{align*}
U := U_s \cap U_u
\end{align*}
after shrinking both stable and unstable constructions so that this intersection is still a product-type coordinate neighbourhood of $p$. Replace $W^s_{\mathrm{loc}}(p)$ and $W^u_{\mathrm{loc}}(p)$ by their connected components through $p$ inside $U$. The preceding maximality statements remain valid for this smaller isolating neighbourhood by applying the graph transform construction with the corresponding smaller product ball.
Let $d$ be a Riemannian distance defined on a neighbourhood of $p$. Shrink $U$ so that $\overline{U}$ lies in the coordinate chart and in the domain on which $d$ is defined. Since the coordinate map $\varphi$ is a $C^r$ diffeomorphism onto its image and the Riemannian metric is continuous, there exist constants $m>0$ and $M_0>0$ such that
\begin{align*}
m|\varphi(x)-\varphi(y)| \le d(x,y) \le M_0|\varphi(x)-\varphi(y)|
\end{align*}
for all $x,y\in U$.
Let $x\in W^s_{\mathrm{loc}}(p)$ and define
\begin{align*}
z:=\varphi(x)\in \Gamma_s.
\end{align*}
Since $\varphi(p)=0$ and $\varphi(f^n(x))=F^n(z)$ for every $n\in\mathbb{N}_0$ for which the orbit remains in $U$, the coordinate contraction estimate gives
\begin{align*}
d(f^n(x),p) \le M_0 |F^n(z)|.
\end{align*}
Using the stable graph estimate,
\begin{align*}
d(f^n(x),p) \le M_0K_s\theta_s^n |z|.
\end{align*}
Using the lower metric comparison with $y=p$,
\begin{align*}
|z|=|\varphi(x)-\varphi(p)| \le m^{-1}d(x,p).
\end{align*}
Therefore
\begin{align*}
d(f^n(x),p) \le \frac{M_0K_s}{m}\theta_s^n d(x,p).
\end{align*}
The same argument applied to $f^{-1}$ gives, for every $x\in W^u_{\mathrm{loc}}(p)$ and every $n\in\mathbb{N}_0$,
\begin{align*}
d(f^{-n}(x),p) \le \frac{M_0K_u}{m}\theta_u^n d(x,p).
\end{align*}
Define
\begin{align*}
C := \max\left\{\frac{M_0K_s}{m},\frac{M_0K_u}{m}\right\}
\end{align*}
and
\begin{align*}
\theta := \max\{\theta_s,\theta_u\}.
\end{align*}
Since $\theta_s,\theta_u\in(0,1)$, one has $\theta\in(0,1)$. The two displayed estimates become exactly the asserted stable and unstable exponential estimates. This completes the construction of the local stable and unstable disks, their tangency properties, their local invariance, their maximal local characterisations, and their exponential contraction estimates.
[/step]
