# Diffeomorphisms with Shadowable Measures
## 1 Introduction
We say that \(\mu\in\mathcal{M}(X)\) is a _shadowable measure_ of \(f\) (or simply, \(f\) is \(\mu\)-_shadowable_) if for any \(\epsilon>0\), there exists \(\delta>0\) such that  
\[\mu_{n}(\mathcal{SPO}(\delta,\epsilon,n))=\mu_{n}(\mathcal{PO}(\delta,n))\]  
for any \(n\in\mathbb{N}\) (if \(A\) is a subset of \(X\), then we define the shadowable measure for \(f_{|A}\) by the same manner). Observe that if \(f:X\to X\) has the shadowing property, then \(f\) is \(\mu\)-shadowable for any \(\mu\in\mathcal{M}(X)\). Denote by \(\text{supp}(\mu)\) the support of \(\mu\in\mathcal{M}(X)\). Then, since \(X\) is compact, we can see that if \(f\) is \(\mu\)-shadowable and \(f(\text{supp}(\mu))=\text{supp}(\mu)\), then \(f:\text{supp}(\mu)\to\text{supp}(\mu)\) has the shadowing property (see Lemma 1 in Section 3). Let \(\mathcal{M}\) be a closed \(C^{\infty}\) manifold, and let \(\text{Diff}(M)\) be the space of diffeomorphisms of \(M\) endowed with the \(C^{1}\)-topology. In this paper, we denote by \(\mathcal{S}\) the set of \(f\in\text{Diff}(M)\) possessing the shadowing property. It is well-known that \(f\in\text{int}\mathcal{S}\) if and only if \(f\) satisfies both Axiom A and the strong transversality condition (see [5; 6; 7] among others), where int means the \(C^{1}\)-interior of the set. Hereafter, let \(P(f)\) be the set of periodic points of \(f\in\text{Diff}(M)\), and let \(\Omega(f)\) be the set of non-wandering points of it. Let \(\Lambda\subset M\) be a closed \(f\)-invariant set. Recall that \(\Lambda\) is _hyperbolic_ if the tangent bundle \(T_{\Lambda}M\) has a \(Df\)-invariant splitting \(E^{s}\oplus E^{u}\) with constants \(C>0\) and \(0<\lambda<1\) such that  
\[\|Df^{n}\|_{E^{s}_{x}}\|\leq C\lambda^{n}\ \text{ and }\ \|Df^{-n}\|_{E^{u}_{x}}\|\leq C\lambda^{n}\]  
for all \(x\in\Lambda\) and \(n\geq 0\). It is well-known that if \(\Lambda\) is hyperbolic, then \(f_{|\Lambda}\) has the shadowing property (see [8]). Recall that the shadowing point is not necessarily contained in \(\Lambda\). We say that \(f\) satisfies _Axiom A_ if \(\Omega(f)\) is hyperbolic and \(\Omega(f)=\overline{P(f)}\). The _stable manifold_ of a point \(x\) is the set  
\[W^{s}(x)=\{y\in M:d(f^{n}(x),f^{n}(y))\to 0\text{ as }n\to\infty\}.\]  
The _unstable manifold_\(W^{n}(x)\) of \(x\) is also defined analogously for \(n\to-\infty\). It is well-known that if \(\Lambda\) is hyperbolic, then \(W^{s}(x)\) and \(W^{n}(x)\) are both immersed manifolds for each \(x\in\Lambda\) (for instance, see [8]). Let \(f\) satisfies Axiom A. We say that \(f\) satisfies the strong transversality condition if all the stable and the unstable manifolds are transverse at any point of their intersection. To describe our results, let us introduce some notation on probability measures of \(M\). Let us endow \(\mathcal{M}(M)\) with the weak topology. It is well-known that the set of \(f\)-invariant measures, \(\mathcal{M}_{f}(M)\), is a non-empty compact subset of \(\mathcal{M}(M)\). We say that \(\mu\in\mathcal{M}(M)\) is _atomic_ if there exists a point \(x\in M\) such that \(\mu(\{x\})>0\) (denote by \(\mathcal{A}(M)\) the set of atomic measures of \(M\)). It is known that the set of non-atomic measures is a residual set in \(\mathcal{M}(M)\) (see [9]). We define two subsets \(\mathcal{PS}\) and \(\mathcal{PLS}\) of \(\text{Diff}(M)\) as follows from a measure-theoretical view point and characterize them in the context of geometric theory of differentiable dynamical systems. \[\mathcal{PS}=\{f\in\text{Diff}(M):f\text{ is $\mu$-shadowable for any }\mu\in\mathcal{M}(M)\setminus\mathcal{A}(M)\},\]  
\(\mathcal{PLS}=\{f\in\text{Diff}(M):f\text{ is $\mu$-shadowable for any }\mu\in\mathcal{M}_{f}(M)\setminus\mathcal{A}(M)\},\)  
Recall that if \(f\in\mathcal{S}\), then \(f\) is \(\mu\)-shadowable for any \(\mu\in\mathcal{M}(M)\setminus\mathcal{A}(M)\). Thus  
\[\mathcal{S}\subset\mathcal{PS}\subset\mathcal{PLS}.\]  
In this paper, we prove the following two theorems. **Theorem 1**.