# Diffeomorphisms with Shadowable Measures
## 4 Proof of Theorem 2
Then  
\[T_{p}M=E_{p}^{s}\oplus E_{p}^{c}\oplus E_{p}^{u}.\]  
For \(x\in B_{\epsilon_{0}}(p)\) we define  
\[\mathcal{L}_{x}^{su}=\exp_{p}\left(\exp_{p}^{-1}(x)+E_{p}^{s}\oplus E_{p}^{u} \right)\cap B_{\epsilon_{0}}(p).\]  
If we put \(\epsilon_{2}=\tau\cdot\frac{\epsilon_{1}}{4}\) for \(0<\tau<1\), then, we may assume  
\[\epsilon_{2}<\min\left\{d(y,z)\ \left|\ y\in\bigcup_{x\in B_{\epsilon_{2}}(p)} \mathcal{L}_{x}^{su},\,z\in\mathcal{L}_{p}\setminus B_{\frac{\epsilon_{1}}{4} }(p)\right.\right\}\ (<\frac{\epsilon_{1}}{4})\]  
by reducing \(\tau\) if necessary. Let \(m_{\mathcal{L}_{p}}\) be a normalized Lebesgue measure on \(\mathcal{L}_{p}\). We define \(\mu\in\mathcal{M}_{\varphi}(M)\) by  
\[\mu(C)=\frac{1}{\pi(p)}\sum_{j=0}^{\pi(p)-1}m_{\mathcal{L}_{p}}\left(\varphi^ {-j}\left(C\cap\varphi^{j}(\mathcal{L}_{p})\right)\right)\]  
for any Borel set \(\mathcal{C}\) of \(M\). Observe that \(\mu\in\mathcal{M}_{\varphi}(M)\setminus\mathcal{A}(M)\). Let \(\delta>0\). By the continuity of \(\varphi\), there exists \(0<\delta_{1}<\epsilon_{2}\) such that \(d(x,y)<\frac{\delta_{1}}{2}\) implies  
\[d(\varphi^{i}(x),\varphi^{i}(y))<\frac{\delta}{2}\]  
for \(x,y\in M\) and \(0\leq i\leq\pi(p)-1\). Choose an integer \(m>0\) such that \(\frac{\epsilon_{1}}{2}<m\delta_{1}<\frac{3\epsilon_{1}}{4}\). Then there are points  
\[x_{0},x_{1},x_{2},\cdots,x_{m-1}\in\mathcal{L}_{p}\]  
such that  
(i) \(x_{0}=p\),  
(ii) \(B_{\frac{\delta_{1}}{2}}(x_{k})\cap B_{\frac{\delta_{1}}{2}}(x_{k+1})\
eq \emptyset\) for \(0\leq k\leq m-2\),  
(iii) \(d(x_{0},x_{k})<\frac{3\epsilon_{1}}{4}\ \ (k=1,2,\cdots,m-1)\), \(d(x_{0},x_{m-1})>\frac{3\epsilon_{1}}{8}\). For any \(0\leq k\leq m-1\) and \(0\leq\ell\leq\pi(p)-1)\), we set  
\[I_{k\pi(p)+\ell}=\varphi^{\ell}(B_{\frac{\delta_{1}}{2}}(x_{k})\cap\mathcal{L} _{p}).\]  
The set \(S_{m}\) is defined by  
\[S_{m}=\{\{y_{i}\}\in M^{m\pi(p)}\mid y_{i}\in I_{i},\ 0\leq i\leq m\pi(p)-1\}.\]Then it is easily checked that \(S_{m}\subset\mathcal{PO}(\delta,m\pi(p))\) and \(\mu_{m\pi(p)}(S_{m})>0\). We show that  
\[\mathcal{SPO}\left(\delta,\frac{\epsilon_{2}}{2},m\pi(p)\right)\cap S_{m}= \emptyset.\]  
Let \(\{y_{i}\}\in S_{m}\) and \(y\in M\) satisfy \(d(\varphi^{i}(y),y_{i})<\frac{\epsilon_{2}}{2}\) for \(0\leq i\leq m\pi(p)-1\). Since  
\[d(\varphi^{k\pi(p)}(y),p) \leq d(\varphi^{k\pi(p)}(y),y_{k\pi(p)})+d(y_{k\pi(p)},x_{k\pi(p)})+d (x_{k\pi(p)},p)\] \[\mathcal{L}_{x}^{su}=\exp_{p}\left(\exp_{p}^{-1}(x)+E_{p}^{s}\oplus E_{p}^{u} \right)\cap B_{\epsilon_{0}}(p).\]0  
we have \(d(\varphi^{k\pi(p)+j}(y),\varphi^{i}(p))\leq\epsilon_{0}\) for \(0\leq k\leq m-1\) and \(0\leq j\leq\pi(p)-1\). If we put \(\exp_{p}^{-1}(y)=v^{s}+v^{\varepsilon}+v^{u},\ v^{\varepsilon}\in E_{p}^{ \varepsilon}\ (\sigma=s,c,u)\), then we have  
\[\mathcal{L}_{x}^{su}=\exp_{p}\left(\exp_{p}^{-1}(x)+E_{p}^{s}\oplus E_{p}^{u} \right)\cap B_{\epsilon_{0}}(p).\]1 \[\mathcal{L}_{x}^{su}=\exp_{p}\left(\exp_{p}^{-1}(x)+E_{p}^{s}\oplus E_{p}^{u} \right)\cap B_{\epsilon_{0}}(p).\]2 \[\mathcal{L}_{x}^{su}=\exp_{p}\left(\exp_{p}^{-1}(x)+E_{p}^{s}\oplus E_{p}^{u} \right)\cap B_{\epsilon_{0}}(p).\]3 \[\mathcal{L}_{x}^{su}=\exp_{p}\left(\exp_{p}^{-1}(x)+E_{p}^{s}\oplus E_{p}^{u} \right)\cap B_{\epsilon_{0}}(p).\]4  
Since  
\[\mathcal{L}_{x}^{su}=\exp_{p}\left(\exp_{p}^{-1}(x)+E_{p}^{s}\oplus E_{p}^{u} \right)\cap B_{\epsilon_{0}}(p).\]5  
we have \(d(\varphi^{(m-1)\pi(p)}(y),y_{(m-1)\pi(p)})\geq\epsilon_{2}\), which is a contradiction. Therefore we have \(\mathcal{SPO}(\delta,\frac{\epsilon_{2}}{2},m\pi(p))\cap S_{m}=\emptyset\). But this contradicts with \(\varphi\in\mathcal{U}(f)\). _Case 2_. \(\dim E_{p}^{c}=2\). In the proof of the second case, to avoid notational complexity, we consider only the case \(g(p)=p\). As in the first case, by Lemma 2, there are \(\epsilon_{0}>0\) and \(\varphi\in\mathcal{U}(f)\) such that \(\varphi(p)=g(p)=p\) and  
\[\mathcal{L}_{x}^{su}=\exp_{p}\left(\exp_{p}^{-1}(x)+E_{p}^{s}\oplus E_{p}^{u} \right)\cap B_{\epsilon_{0}}(p).\]6  
if \(x\in B_{\epsilon_{0}}(p)\).