# Diffeomorphisms with Shadowable Measures
## 4 Proof of Theorem 2  
In what follows, we give a proof of Theorem 2. Recall that \(f\in\mathcal{F}^{1}(M)\) if and only if \(f\) satisfies both Axiom A and no-cycle condition by [10]. Thus, to get the conclusion, it is enough to show that \(\mathsf{int}\mathcal{PLS}\subset\mathcal{F}^{1}(M)\). Suppose \(f\in\mathsf{int}\mathcal{PLS}\). The existence of the non-hyperbolic periodic point of \(f\) easily gives a small periodic curve with period \(l\) for some \(\varphi\)\(C^{1}\)-nearby \(f\) such that the restriction of the map \(\varphi^{l}\) to the curve is the identity map (see End of the proof of Theorem 2 below). From this, we will show that periodic points of any \(g\), \(C^{1}\)-nearby \(f\), are all hyperbolic. This kind of result was first obtained by the first author in ([12], Proposition 1). To show this fact we use the next lemma several times. **Lemma 2**. : _Let \(f\in\mathrm{Diff}(M)\) and let \(\mathcal{U}(f)\) be given. Then there is \(\delta>0\) such that for a finite set \(\{x_{1},x_{2},\cdots,x_{N}\}\), a neighborhood \(U\) of \(\{x_{1},x_{2},\cdots,x_{N}\}\) and linear maps \(L_{i}:T_{x_{i}}M\to T_{f(x_{i})}M\) satisfying \(\|L_{i}-D_{x_{i}}f\|\leq\delta\) for all \(1\leq i\leq N\), there are \(\epsilon_{0}>0\) and \(g\in\mathcal{U}(f)\) such that_  
1. \(g(x)=f(x)\) _if_ \(x\in M\setminus U\)_, and_
2. \(g(x)=\exp_{f(x_{i})}\circ L_{i}\circ\exp_{x_{i}}^{-1}(x)\) _if_ \(x\in B_{\epsilon_{0}}(x_{i})\) _for all_ \(1\leq i\leq N\)_._  
Observe that the assertion \((b)\) implies that  
\[g(x)=f(x)\text{ if }x\in\{x_{1},x_{2},\cdots,x_{N}\}\]  
and that \(D_{x_{i}}g=L_{i}\) for all \(1\leq i\leq N\). The proof is essentially contained in the proof of ([13], Lemma 1). **End of the proof of Theorem 2**. Suppose \(f\in\mathsf{int}(\mathcal{PLS})\), and we show that \(f\in\mathcal{F}^{1}(M)\). By contradiction, assume that \(f\
otin\mathcal{F}^{1}(M)\) and we shall derive a contradiction. Since \(f\in\mathsf{int}(\mathcal{PLS})\), there exists a \(C^{1}\)-neighborhood \(\mathcal{U}(f)\) of \(f\) such that for any \(g\in\mathcal{U}(f)\) and any \(\mu\in\mathcal{M}_{g}(M)\setminus\mathcal{A}(M)\), \(g\) is \(\mu\)-shadowable. On the other hand, it follows from the assumption \((f\
otin\mathcal{F}^{1}(M))\) that there are \(g\in\mathcal{U}(f)\) and non-hyperbolic periodic point \(p\) of \(g\). At first, by Lemma 2, with a small modification of the map \(g\) with respect to the \(C^{1}\)-topology, we may assume that \(D_{p}g^{\pi(p)}\) has only one eigenvalue \(\lambda\) with modulus equal to \(1\) (or only one pair of complex conjugated eigenvalues, see Case 2 below). Denote by \(E_{p}^{c}\) the eigenspace corresponding to \(\lambda\). _Case 1_. \(\dim E_{p}^{c}=1\). In this case, we suppose further that \(\lambda=1\) for simplicity (the other case is similar). Then, by Lemma 2, there are \(\epsilon_{0}>0\) and \(\varphi\in\mathcal{U}(f)\) such that \(\varphi^{\pi(p)}(p)=g^{\pi(p)}(p)=p\) and  
\[\varphi(x)=\exp_{g^{i+1}(p)}\circ D_  
Here \(E_{p}^{c}(\epsilon_{1})\) is the \(\epsilon_{1}\)-ball in \(E_{p}^{c}\) centered at the origin \(O_{p}\). Since the eigenvalue \(\lambda\) of \(D_{p}g^{\pi(p)}|_{E_{p}^{c}}\) is \(1\), we have  
\[ \begin{array}{l}\varphi^{i}(\mathcal{L}_{p})\cap\varphi^{j}(\mathcal{L}_{p})= \emptyset\ \ (0\leq i\
eq j\leq\pi(p)-1),\\ \varphi^{\pi(p)}(\mathcal{L}_{p})=\mathcal{L}_{p},\text{ and }\\ \varphi^{\pi(p)}|_{\mathcal{L}_{p}}\text{ is the identity map. }\end{array} \]  
Denote by \(E_{p}^{s}\) the eigenspace corresponding to the eigenvalues of \(D_{p}g^{\pi(p)}\) with modulus less than \(1\), and by \(E_{p}^{u}\) the eigenspace corresponding to the eigenvalues with modulus more than \(1\).