# Diffeomorphisms with Shadowable Measures
## 1 Introduction
Observe that for any \(\delta>0\) and \(n\in\mathbb{N}\), we can construct a \(\delta\)-pseudo-orbit \(\{x_{i}\}_{i=0}^{n-1}\) of \(g\) starting at a neighborhood of \(p_{1}\), passing through \(I\), and arriving at a neighborhood of \(p_{4}\). It is easy to see that there exists \(\epsilon>0\) such that for any \(\delta>0\), there exists this kind of \(\delta\)-pseudo-orbit \(\{x_{i}\}_{i=0}^{n-1}\) which cannot be \(\epsilon\)-shadowed by any orbit of \(g\). Since \(\
u(U)=0\), the \(\
u_{n}\)-measure of the set of such a kind of \(\delta\)-pseudo-orbits is \(0\). On the other hand, since \(g\) is, in a sense, hyperbolic in the outside of \(U\), we can see that any pseudo-orbit can be shadowed by an orbit of \(g\). For, since \(p_{4}\) is a sink fixed point, there is a neighbourhood \(W(p_{4})\) of \(p_{4}\) and \(0<\eta<1\) such that for any \(x,y\in W(p_{4})\), \(d(g(x),g(y))\leq\eta d(x,y)\). For any \(\epsilon>0\), choose \(\delta>0\) small enough so that \(\delta<e(1-\eta)\). Let \(\{x_{i}\}_{i=0}^{n-1}\) be a \(\delta\)-pseudo-orbit of \(g\) contained in \(W(p_{4})\). If \(d(y,x_{0})<\epsilon\) for some \(y\in M\), then  
\[\|Df^{n}\|_{E^{s}_{x}}\|\leq C\lambda^{n}\ \text{ and }\ \|Df^{-n}\|_{E^{u}_{x}}\|\leq C\lambda^{n}\]5  
by the choice of \(\delta\). Moreover, since \(d(g(x_{1}),x_{2})<\delta\), we have  
\[\|Df^{n}\|_{E^{s}_{x}}\|\leq C\lambda^{n}\ \text{ and }\ \|Df^{-n}\|_{E^{u}_{x}}\|\leq C\lambda^{n}\]6  
By continuing this manner, we can see that \(d(g^{i}(y),x_{i})<\epsilon\) for any \(0\leq i\leq n-1\). Thus, we have the following. **Claim 1**. _For any \(\epsilon>0\), there is \(\delta>0\) such that any \(\delta\)-pseudo-orbit \(\{x_{i}\}_{i=0}^{n-1}\) of \(g\) which is contained in \(W(p_{4})\) can be \(\epsilon\)-shadowed by some point._  
Observe that since \(p_{1}\) is a source fixed point, a similar property also holds in a neighbourhood \(W(p_{1})\) of \(p_{1}\) with respect to \(g^{-1}\). The assertion of the following claim can be easily checked. \[\|Df^{n}\|_{E^{s}_{x}}\|\leq C\lambda^{n}\ \text{ and }\ \|Df^{-n}\|_{E^{u}_{x}}\|\leq C\lambda^{n}\]7 The structure of the stable and unstable manifolds. **Claim 2**. _There are a constant \(N>0\) and \(\delta_{0}>0\) such that for any \(\delta\)-pseudo-orbit \(\{x_{i}\}_{i=0}^{n-1}\) of \(g\ (0<\delta<\delta_{0})\), if_  
\[\|Df^{n}\|_{E^{s}_{x}}\|\leq C\lambda^{n}\ \text{ and }\ \|Df^{-n}\|_{E^{u}_{x}}\|\leq C\lambda^{n}\]8  
_then \(n\leq N\)._  
It is not hard to show that for any \(\epsilon>0\), there exists \(0<\delta_{1}<\delta_{0}\) such that any \(\delta\)-pseudo-orbit \(\{x_{i}\}_{i=0}^{n-1}\) of \(g\ (0<\delta<\delta_{1})\) with \(n\leq N\) can be \(\epsilon\)-shadowed some point. Combining this fact with the assertion of Claim 1, we can see that the map \(g\) has the shadowing property in the outside of \(U\). Thus \(\
u\) is a shadowable measure. **3. Proof of Theorem 1**  
In this section, we give a proof of Theorem A. Let \(f\in\mbox{Diff}(M)\). As we explained before, it has shown that \(f\in\mbox{int}S\) if and only if \(f\) satisfies both Axiom A and the strong transversality condition. Thus, to get the conclusion, it is enough to show that \(\mbox{int}\mathcal{PS}\subset\mbox{int}\mathcal{S}\). In the following lemma, let \((X,d)\) be a compact metric space, and let \(f:X\to X\) be a homeomorphism. **Lemma 1**. : _Let \(\mu\in\mathcal{M}(X)\). If \(f\) is \(\mu\)-shadowable, then for any \(\epsilon>0\), there exist \(\delta>0\) such that for any \(n\geq 1\) and \(\delta\)-pseudo-orbit \(\{x_{i}\}_{i=0}^{n-1}\subset supp(\mu)\) of \(f\), there is \(y\in X\)\(\epsilon\)-shadowing the pseudo-orbit._  
Proof. : Since \(\mu\) is shadowable, for any \(\epsilon>0\), there exists a \(\delta>0\) such that for any \(n\in\mathbb{N}\),  
\[\|Df^{n}\|_{E^{s}_{x}}\|\leq C\lambda^{n}\ \text{ and }\ \|Df^{-n}\|_{E^{u}_{x}}\|\leq C\lambda^{n}\]9  
Fix \(n\) and let a \(\delta\)-pseudo-orbit \(\{x_{i}\}_{i=0}^{n-1}\subset\mbox{supp}(\mu)\) of \(f\) be given.