# Diffeomorphisms with Shadowable Measures
## 1 Introduction
It is easy to see that the pseudo-orbit cannot be \(\frac{1}{3}\)-shadowed by \(id_{S^{1}}\)-orbit for any \(y\in S^{1}\). Observe that if \(|\theta_{1}-\theta_{2}|\) is small enough, then  
\[\|Df^{n}\|_{E^{s}_{x}}\|\leq C\lambda^{n}\ \text{ and }\ \|Df^{-n}\|_{E^{u}_{x}}\|\leq C\lambda^{n}\]2  
Thus, it is not hard to show that the rotation map does not have the shadowing property as in the case \(id_{S^{1}}\). Let \(\mu\) be the usual Lebesgue probability measure on \(S^{1}\) so that \(\mu\) is \(\rho_{\alpha}\)-invariant. Then, as in the case of the identity map of the unit interval \(I\), the measure of the set of non-shadowable pseudo-orbits is positive. As stated before, the last example does not have the shadowing property, but has a shadowable measure. For the description, we need some notations. Let \(M^{2}\) be a surface, and let \(f\in\operatorname{Diff}(M^{2})\) satisfy Axiom A. The so-called \(C^{0}\)-transversality condition for \(f\) is introduced by [11]. Roughly speaking, we say that the stable manifold \(W^{s}(z)\) and the unstable manifold \(W^{u}(z)\) of \(z\in M^{2}\) are \(C^{0}\)-transverse at \(z\) if these sets cross at \(z\) as the graph of the function \(y=x^{3}\) crosses the \(x\)-axis at the origin in the \(xy\)-plane. We say that \(f\) satisfies the \(C^{0}\)-transversality condition if all the stable and the unstable manifolds are \(C^{0}\)-transverse at any point of their intersection. It is proved therein that \(f\) satisfies the \(C^{0}\)-transversality condition if and only if \(f\) has the shadowing property. **Example 3**. : _(Axiom A diffeomorphisms with no-cycles)._  
We represent two-torus \(\mathbb{T}^{2}\) as the square \([-2,2]\times[-2,2]\) with identified opposite sides in the Euclidean space \(\mathbb{R}^{2}\). Let \(g:\mathbb{T}^{2}\to\mathbb{T}^{2}\) be a diffeomorphism with the following properties (see Figure 1):  
\((\mathbf{1})\): the non-wandering set \(\Omega(g)\) of \(g\) is the union of \(4\) hyperbolic fixed points, that is, \(\Omega(g)=\{p_{1},\ p_{2},\ p_{3},\ p_{4}\}\), where \(p_{1}\) is a source, \(p_{4}\) is a sink, and \(p_{2},\ p_{3}\) are saddles;  
\((\mathbf{2})\): with respect to coordinates \((v,w)\in[-2,2]\times[-2,2]\), the following conditions hold:  
\((\ref{eq:1})\ p_{1}=(1,2),\ p_{2}=(1,0),\ p_{3}=(-1,0),\ p_{4}=(-1,2)\),  
\((\ref{eq:2})\ W^{u}(p_{2})\cup\{p_{3}\}=W^{s}(p_{3})\cup\{p_{2}\}=[-2,2] \times\{0\}\),  
\[\|Df^{n}\|_{E^{s}_{x}}\|\leq C\lambda^{n}\ \text{ and }\ \|Df^{-n}\|_{E^{u}_{x}}\|\leq C\lambda^{n}\]3  
where \(W^{s}(p_{i})\) and \(W^{u}(p_{i})\) are the stable and unstable manifolds \((i=2,3)\), respectively, defined as usual;  
\((\ref{eq:2})\): there exist neighborhoods \(V_{2},\ V_{3}\) of \(p_{2},\ p_{3}\) such that  
\[\|Df^{n}\|_{E^{s}_{x}}\|\leq C\lambda^{n}\ \text{ and }\ \|Df^{-n}\|_{E^{u}_{x}}\|\leq C\lambda^{n}\]4(2.4) there exists a neighborhood \(V\) of the point \(O=(0,0)\) such that \(g(V)\subset V_{3}\), \(g^{-1}(V)\subset V_{2}\) and \(g^{-1}\) is affine on \(g(V)\),  
(2.5) the eigenvalues of \(D_{p_{3}}g\) are \(\mu\), \(\
u\) with \(\mu>1\), \(0<\
u<1\), and the eigenvalues of \(D_{p_{2}}g\) are \(\lambda\), \(\kappa\) with \(0<\lambda<1\), \(\kappa>1\). Please note that \(g\) satisfies both Axiom A and the no-cycle condition (i.e., it is \(\Omega\)-stable) but does not have the shadowing property. Indeed, since \(W^{s}(p_{3})\setminus\{p_{3}\}=W^{u}(p_{2})\setminus\{p_{2}\}\), the stable manifold and the unstable manifold are not \(C^{0}\)-transverse, and thus \(g\) does not have the shadowing property. However, we show herewith that there exists shadowable measure for \(g\). Indeed, put \(I=[-2,2]\times\{0\}\) and let \(U\) be a small neighborhood of \(I\) (see Figure 1). Define \(\
u\) as the Lebesgue measure on \(\mathbb{T}^{2}\setminus U\) such that \(\
u(\mathbb{T}^{2})=1\) and \(\
u(U)=0\).