# Diffeomorphisms with Shadowable Measures
## 1 Introduction  
The notion of pseudo-orbits very often appears in the several branches of modern theory of dynamical systems, and, especially, the pseudo-orbit shadowing property usually plays an important role in the investigation of stability theory and ergodic theory as well as expansivity. Let \((X,d)\) be a compact metric space, and let \(f:X\to X\) be a homeomorphism. For \(\delta>0\), a sequence of points \(\{x_{i}\}_{i=a}^{b}\subset X\left(-\infty\leq a<b\leq\infty\right)\) is called a \(\delta\)_-pseudo-orbit_ of \(f\) if \(d(f(x_{i}),x_{i+1})<\delta\) for all \(a\leq i\leq b-1\). Denote by \(f_{|A}\) the restriction of \(f\) to a subset \(A\) of \(X\). Let \(\Lambda\subset X\) be a closed \(f\)-invariant set, that is, \(f(\Lambda)=\Lambda\). We say that \(f_{|\Lambda}\) has the _shadowing property_ if for every \(\epsilon>0\), there is \(\delta>0\) such that for any \(n\in\mathbb{N}\) and \(\delta\)-pseudo-orbit \(\{x_{i}\}_{i=0}^{n-1}\subset\Lambda\) of \(f\), there is \(y\in X\,\epsilon\)-shadowing the pseudo-orbit, that is, \(d(f^{i}(y),x_{i})<\epsilon\) for all \(0\leq i\leq n-1\). Notice that only \(\delta\)-pseudo-orbits of \(f\) \"contained in \(\Lambda\)\" can be \(\epsilon\)-shadowed, but shadowing point \(y\in X\) is \"not necessarily\" contained in \(\Lambda\). We say that \(f\) has the _shadowing property_ if \(X=\Lambda\) in the above definition. Since \(X\) is compact, it is not hard to show that if \(f_{|\Lambda}\) has the shadowing property, then every pseudo-orbit \(\{x_{i}\}_{i=-\infty}^{\infty}\subset\Lambda\) can be shadowed by some true orbit. The notion of expansivity has been intensively studied by several researchers, mainly from the topological view point, and lots of important fruitful results were obtained. Nowadays, expansive theory of dynamical systems has been well developed in both of the geometric theory and ergodic theory of dynamical systems as well as that of shadowing theory. Recently, the notion of the expansive measures was introduced by [PERSON] and [PERSON] [1] as a generalization of the notion of expansivity from the measure theoretical view point, and they exhibit the effectiveness of the use of it in expansive theory of dynamical systems (see [2; 3; 4] among others). In this paper, inspired by the work of [PERSON] and [PERSON], we introduce the notion of shadowable measures for a homeomorphism of a compact metric space as a generalization of the shadowing property from the measure theoretical view point, and consider the set of diffeomorphisms of a closed differentiable manifolds possessing the shadowable measures. The notion of such kind of measures is recently introduced by [PERSON] and [PERSON] [2], and they consider the shadowability of pseudo-orbits of a dynamical system from the measure theoretical view point. In our opinion, however, the introduced measure does not properly reflect the behavior of pseudo-orbits of the system. In fact, for a givenpseudo-orbit \(\{x_{i}\}_{i=0}^{\infty}\) of the system, they measure only the initial point \(x_{0}\) of the pseudo-orbit, whole of the pseudo-orbit is not considered (see ([2], Definition 2.5)). Recall that \((X,d)\) is a compact metric space and \(f:X\to X\) is a homeomorphism of \(X\). Let \(X^{n}=X\times\cdots\times X\) (the \(n\)-times of direct product) be the sequences of points of \(X\) with length \(n\) (\(n\in\mathbb{N}\)), and denote by \(\mathcal{M}(X)\) the space of Borel probability measures of \(X\). For any \(\mu\in\mathcal{M}(X)\) (which is not necessarily \(f\)-invariant), let \(\mu_{n}=\mu\times\cdots\times\mu\) (\(n\)-times) be the direct product measure of \(X^{n}\). For any \(\delta>0\), denote by \(\mathcal{PO}(\delta,n)\) the space of \(\delta\)-pseudo-orbits \(\{x_{i}\}_{i=0}^{n-1}\in X^{n}\) of \(f\), and for \(\epsilon>0\), denote by \(\mathcal{SPO}(\delta,\epsilon,n)\) (\(\subset\mathcal{PO}(\delta,n)\)) the set of \(\delta\)-pseudo-orbits \(\epsilon\)-shadowed by some point.