In this paper, given a homeomorphism \(f\) of a compact metric space \(X\), we show that the set of all asymptotic average shadowable points of \(f\) is an open and invariant set and \(f\) has the asymptotic average shadowing property if and only if the set of all asymptotic average shadowable points of \(f\) is \(X\) if and only if any Borel probability measure \(\mu\) of \(X\) has the asymptotic average shadowing property.
The asymptotic average shadowing property was introduced by Gu [2]. Gu proved in [2] that if a homomorphism \(f\) of a compact metric space \(X\) has the asymptotic average shadowing property, then it is chain transitive. Honary and Bahabadi [3] proved that if a diffeomorphism \(f\) of a compact smooth manifold \(M\) has the asymptotic average shadowing property and \(M\) is not finite and \({\rm dim}M=2\), then the \(C^1\) interior of the set of all \(C^1\) diffeomorphisms with the asymptotic average shadowing property is characterized by the set of \(\Omega\) stable diffeomorphisms, and Lee [7] proved that if \({\rm dim}M\geq 2\) then the \(C^1\) interior of the set of all \(C^1\) diffeomorphisms with the asymptotic average shadowing property is characterized by the set of weak hyperbolic (dominated splitting) diffeomorphisms.
Also Lee [8] proved that for \(C^1\) generic diffeomorphism \(f\) of a compact smooth manifold \(M\) with \({\rm dim}M\geq2\) then \(f\) is weakly hyperbolic. Thus, the asymptotic mean shadowing property is an interesting topic to study dynamical systems (topological and smooth dynamical systems). Regarding the various shadowing points, Morales [9] was the first to introduce the shadowable points. After his research, many valuable results are published in [1, 5, 6, 4] using the various shadowable points. The asymptotic average shadowable point was introduced by Rego and Arbieto [10]. As the notion of [10], we will study in the paper. More specifically, given a homeomorphism \(f\) of a compact metric space \(X\), we prove that the set of all asymptotic average shadowable points of \(f\) is an open and invariant set and \(f\) has the asymptotic average shadowing property if and only if the set of all asymptotic average shadowable points of \(f\) is \(X\) if and only if any Borel probability measure \(\mu\) of \(X\) has the asymptotic average shadowing property.
Let \((X, d)\) be a compact metric space with metric \(d\) and let \(f:X\to X\) be a homeomorphism. For any \(\delta>0\), An infinite sequence \(\{x_i:i\in\mathbb{Z}\}\) is said to be \(\delta\)-pseudo-orbit of \(f\) if \(d(f(x_i), x_{i+1})<\delta\) for all \(i\in \mathbb{Z}.\) We say that \(f\) has the shadowing property if for every \(\epsilon>0\) there exists \(\delta>0\) such that every \(\delta\) pseudo-orbit \(\{x_i:i\in \mathbb{Z}\}\) can be \(\epsilon\) shadowed by some point in \(X\), i.e. there exists \(y\in X\) such that \(d(f^i(y), x_i)<\epsilon\) for all \(i\in \mathbb{Z}.\) Then the point \(y\in X\) is called the shadowable point of \(f.\) Let \(Sh(f)\) be the set of all shadowable points of \(f\). Morales [9] proved that \(Sh(f)\) is an invariant set and \(f\) has the shadowing property if and only if \(Sh(f)=X.\)
Now we are concerned with a kind of shadowing property and a kind of shadowable points of \(f\). It is a different notion of the shadowing property.
An infinite sequence \(\{x_i:i\in \mathbb{Z}\}\) is said to be asymptotic pseudo-orbit of \(f\) if \[\lim_{n\to\infty} \frac{1}{n}\sum_{i=0}^{n-1}d(f(x_i), x_{i+1})=0, \ \mbox{and}\ \lim_{n\to\infty} \frac{1}{n}\sum_{i=-n}^{-1}d(f(x_i), x_{i+1})=0.\]
A homeomorphism \(f\) of \(X\) has the asymptotic average shadowing property if any asymptotic average pseudo-orbit \(\{x_i:i\in \mathbb{Z}\}\) can be asymptotic average shadowed by some point in \(X\), i.e. there exists \(y\in X\) such that \[\lim_{n\to\infty} \frac{1}{n}\sum_{i=0}^{n-1}d(f^i(y), x_i)=0, \ \mbox{and}\ \lim_{n\to\infty} \frac{1}{n}\sum_{i=-n}^{-1}d(f^i(y), x_i)=0.\]
Then the point \(y\in X\) is called asymptotic average shadowable point of \(f\). Denote by \(AEVSh(f)\) the set of all asymptotic average shadowable points of \(f\).
We say that \(f\) is chain transitive if, for any \(\delta>0\), there exists a sequence \(\{x_i: i=0, 1, \ldots, n\}\) such that \(d(f(x_i), x_{i+1})<\delta\) for all \(i=0, 1, \ldots, n.\) Gu [2] proved that if \(f\) has the asymptotic average shadowing property, then it is chain transitive. Thus, the asymptotic average shadowing property is not equivalent to the shadowing property (see, [2] Section 5.1).
Lemma 2.1. Let \(f\) be a homeomorphism of \(X\). Then \(AEVSh(f)\) is an open set.
Proof. Assume that \(AEVSh(f)\not=\emptyset.\) Let \(y\in AEVSh(f)\) and any asymptotic average pseudo orbit through \(y\) can be asymptotic average shadowed by some point in \(X\). Take \(\epsilon>0\). Let \(x\in B(y, \epsilon)\) and \(\xi=\{x_i:i\in \mathbb{Z}\}\) be an asymptotic average pseudo orbit through \(x\). We make an infinite sequence \(\zeta=\{y_i:i\in \mathbb{Z}\}\) as follows:
\(y_i=x_i\) for \(i\not=0\) and \(y_i=y\) for \(i=0.\) Then we see that \[\begin{aligned} \sum_{i=0}^{n-1} d(f(y_i), y_{i+1})&=d(f(y), y_1)+d(f(y_1), y_2)+\cdots+d(f(y_{n-1}), y_n)\\&=d(f(y), x_1)+d(f(x_1), x_2)+\cdots+d(f(x_{n-1}), x_n)\\& \leq {\rm diam}X+\sum_{i=0}^{n-1}d(f(x_i), x_{i+1}), \end{aligned}\] and \[\begin{aligned} \sum_{i=-n}^{-1} d(f(y_i), y_{i+1})&=d(f(y_{-n}), y_{-n+1})++d(f(y_{-n+1}), y_{-n+2})+\cdots+d(f(y_{-1}), y)\\&=d(f(x_{-n}), x_{-n+1})+d(f(x_{-n+1}), x_{-n+2})+\cdots+d(f(x_{-1}), y)\\& \leq \sum_{i=-n}^{-1}d(f(x_i), x_{i+1})+ {\rm diam}X. \end{aligned}\]
Since \(\xi\) is an asymptotic average pseudo orbit of \(f\), we have that \[\begin{aligned} \lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}d(f(y_i), y_{i+1})\leq & \lim_{n\to\infty}\frac{1}{n}\Big({\rm diam}X+\sum_{i=0}^{n-1}d(f(x_i), x_{i+1})\Big) \\=& \lim_{n\to\infty}\frac{{\rm diam}X}{n}+\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}d(f(x_i), x_{i+1})=0, \end{aligned}\] and \[\begin{aligned} \lim_{n\to\infty}\frac{1}{n}\sum_{i=-n}^{-1}d(f(y_i), y_{i+1})\leq & \lim_{n\to\infty}\frac{1}{n}\Big(\sum_{i=-n}^{-1}d(f(x_i), x_{i+1})+{\rm diam}X\Big) \\=& \lim_{n\to\infty}\frac{1}{n}\sum_{i=-n}^{-1}d(f(x_i), x_{i+1})+\lim_{n\to\infty}\frac{{\rm diam}X}{n}=0. \end{aligned}\]
Thus \(\zeta\) is an asymptotic pseudo obit of \(f\) through \(y.\) Since \(y\in AEVSh(f)\), there is a point \(z\in X\) such that \[\lim_{n\to\infty} \frac{1}{n}\sum_{i=0}^{n-1}d(f^i(z), y_i)=0, \ \mbox{and}\ \lim_{n\to\infty} \frac{1}{n}\sum_{i=-n}^{-1}d(f^i(z), y_i)=0.\]
Since \(y_i=x_i\) for \(i\not=0\) and \(y_0=y\), we also see that \[\lim_{n\to\infty} \frac{1}{n}\sum_{i=0}^{n-1}d\left(f^i(z), x_i\right)=0, \ \mbox{and}\ \lim_{n\to\infty} \frac{1}{n}\sum_{i=-n}^{-1}d(f^i(z), x_i)=0.\]
Thus the asymptotic pseudo orbit \(\xi\) can be asymptotic average shadowed by \(z\in X\). This means that \(x\in AEVSh(f)\) and so \(B(y, \epsilon)\subset ASEVSh(f).\) It implies that \(ASEVSh(f)\) is an open set. ◻
A set \(\Lambda\subset X\) is an invariant if \(f(\Lambda)\subset\Lambda.\)
Lemma 2.2. Let \(f\) be a homeomorphism of \(X\). Then \(AEVSh(f)\) is an invariant set.
Proof. For any \(x\in AEVSh(f)\), let \(\xi=\{x_i:i\in\mathbb{Z} \}\) be an asymptotic average pseudo orbit of \(f\) through \(f(x)\). Since \(\xi\) is an asymptotic average pseudo orbit of \(f\), we have \[\lim_{n\to\infty}\frac{1}{2n}\sum_{i=-n}^{n-1}d(f(x_i), x_{i+1})=0.\]
Since \(f^{-1}\) is uniformly continuous, take \(\epsilon>0\) such that \(d(a, b)<\epsilon/4(a, b\in X)\) implies \(d(f^{-1}(a), f^{-1}(b))<\epsilon.\) Let \(j=\#\{i\in\mathbb{N}^+: d(f(x_i), x_{i+1})\geq \epsilon/4\}.\) Then we have \[\begin{aligned} \sum_{i=0}^{n-1}d(f(x_i), x_{i+1})&=\sum_{i=0}^{j}d(f(x_i), x_{i+1})+\sum_{i=j+1}^{n-1}d(f(x_i), x_{i+1})\\& <\sum_{i=0}^{j}d(f(x_i), x_{i+1})+\sum_{i=j+1}^{n-1}\frac{\epsilon}{4}. \end{aligned}\]
Consider the sequence \(\eta=\{f^{-1}(x_i):i\in\mathbb{Z}\}\). Since \[\begin{aligned} \sum_{i=0}^{n-1}d(f(x_i), x_{i+1})&<\sum_{i=0}^{j}d(f(x_i), x_{i+1})+\sum_{i=j+1}^{n-1}\frac{\epsilon}{4}, \end{aligned}\] we have \[\begin{aligned} \sum_{i=0}^{n-1}d(f^{-1}(f(x_i)), f^{-1}(x_{i+1}))&=\sum_{i=0}^{j}d(f^{-1}(f(x_i)), f^{-1}(x_{i+1}))+\sum_{i=j+1}^{n-1}d(f^{-1}(f(x_i)), f^{-1}(x_{i+1}))\\&< \sum_{i=0}^{j}{\rm dim}X+\sum_{i=j+1}^{n-1}\epsilon . \end{aligned}\]
Since \[\lim_{n\to\infty} \frac{1}{n}\sum_{i=0}^{n-1}d(f(x_i), x_{i+1})=0,\] we see that \[\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}d(f^{-1}(f(x_i)), f^{-1}(x_{i+1}))=0.\]
Similarly, we can see that \[\lim_{n\to\infty}\frac{1}{n}\sum_{i=-n}^{-1}d(f^{-1}(f(x_i)), f^{-1}(x_{i+1}))=0.\]
Then \(\zeta\) is an asymptotic average pseudo orbit of \(f\) through \(x.\) Since \(x\in AEVSh(f)\), there is \(z\in X\) such that \[\lim_{n\to\infty} \frac{1}{n}\sum_{i=0}^{n-1}d(f^i(z), f^{-1}(x_i))=0, \ \mbox{and}\ \lim_{n\to\infty} \frac{1}{n}\sum_{i=-n}^{-1}d(f^i(z), f^{-1}(x_i))=0.\]
By uniform continuity of \(f\), take \(r>0\) such that \(d(f(a), f(b))<r\) whenever \(d(a, b)<r/4(a, b\in X)\). Let \(l=\#\{i\in\mathbb{N}^+: d(f^i(z), f^{-1}(x_i))\geq r/4\}.\) Then we see that
\[\begin{aligned} \sum_{i=0}^{n-1}d(f^i(z), f^{-1}(x_i))&=\sum_{i=0}^{l}d(f^i(z), f^{-1}(x_i))+\sum_{i=l+1}^{n-1}d(f^i(z), f^{-1}(x_i)). \end{aligned}\] . Then we see that \[\begin{aligned} \sum_{i=0}^{n-1}d(f^i(z), f^{-1}(x_i))&=\sum_{i=0}^{l}d(f^i(z), f^{-1}(x_i))+\sum_{i=l+1}^{n-1}\frac{r}{4} \\&<\sum_{i=0}^l {\rm diam}X+\sum_{i=l+1}^{n-1}\frac{r}{4}. \end{aligned}\]
Thus we have that \[\begin{aligned} \sum_{i=0}^{n-1}d\left(f(f^i(z)), x_i\right)&=\sum_{i=0}^{l}d(f(f^i(z)), x_i)+\sum_{i=l+1}^{n-1}d(f(f^i(z)), x_i)\\&<\sum_{i=0}^{l}d(f(f^i(z)), x_i)+\sum_{i=l+1}^{n-1}r \\&<\sum_{i=0}^l {\rm diam}X+\sum_{i=l+1}^{n-1}r. \end{aligned}\]
Since \(f(f^i(z))=f^i(f(z))\) for all \(i\in \mathbb{Z}\), and \[\lim_{n\to\infty}\frac{1}{n} \sum_{i=0}^{n-1}d(f^i(z), f^{-1}(x_i))=0,\] we have \[\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}d(f^i(f(z)), x_i)=0.\]
Similarly, we can see that \[\lim_{n\to\infty}\frac{1}{n}\sum_{i=-n}^{-1}d(f^i(f(z)), x_i)=0.\]
It implies that \(\xi\) is an asymptotic average shadowed by \(f(z)\). Hence \(f(x)\in AEVSh(f),\) and so \(AEVSh(f)\) is an invariant set. ◻
Lemma 2.3. Let \(f\) be a homeomorphism of \(X\). If \(AEVSh(f)=X\), then \(f\) has the asymptotic average shadowing property.
Proof. Suppose that for each \(k\in\mathbb{N}\) there is an asymptotic average pseudo orbit \(\xi^k=\{x_i^k:i\in \mathbb{Z}\}\) which can not be asymptotic average shadowed by a point in \(X.\) Since \(X\) is compact, we assume that \(x_0^k\to p\in X\) as \(k\to\infty.\) Since \(p\in AEVSh(f)=X\), there is an asymptotic average pseudo orbit through \(p\) can be asymptotic average shadowed by a point in \(X\). For any small \(\epsilon>0\), there is \(l\in\mathbb{N}\) such that \(d(x_0^l, p)<\epsilon\) and \(d(f(x_0^l), f(p))<\epsilon.\) We construct a sequence \(\zeta=\{y_i:i\in \mathbb{Z}\}\) as follows: \(y_i=x_i^l\) if \(i\not=0\) and \(y_0=p\) if \(i=0.\) Then we have that
\[\begin{aligned} \sum_{i=0}^{n-1} d(f(y_i), y_{i+1})&=d(f(p), y_1)+d(f(y_1), y_2)+\cdots+d(f(y_{n-1}), y_n)\\&=d(f(p), x^l_1)+d(f(x^l_1), x^l_2)+\cdots+d(f(x^l_{n-1}), x^l_n)\\&\leq d(f(p), f(x_0^l))+d(f(x_0^l), x_1^l)+ +d(f(x^l_1), x^l_2)+\cdots+d(f(x^l_{n-1}), x^l_n) \\& < \epsilon+ {\rm diam}X+\sum_{i=0}^{n-1}d(f(x^l_i), x^l_{i+1}), \end{aligned}\] and \[\begin{aligned} \sum_{i=-n}^{-1} d(f(y_i), y_{i+1})&=d(f(y_{-n}), y_{-n+1})++d(f(y_{-n+1}), y_{-n+2})+\cdots+d(f(y_{-1}), y(=p))\\&=d(f(x^l_{-n}), x^l_{-n+1})+d(f(x^l_{-n+1}), x^l_{-n+2})+\cdots+d(f(x^l_{-1}), p)\\& \leq d(f(x^l_{-n}), x^l_{-n+1})+d(f(x^l_{-n+1}), x^l_{-n+2})+\cdots+d(f(x_{-1}^l), x_0^l)+d(x_0^l, p)\\& < \sum_{i=-n}^{-1}d(f(x^l_i), x^l_{i+1})+ {\rm diam}X+\epsilon. \end{aligned}\]
Thus we see that \[\begin{aligned} \lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1} d(f(y_i), y_{i+1}) < \lim_{n\to\infty}\frac{1}{n}\Big( \epsilon+ {\rm diam}X+\sum_{i=0}^{n-1}d(f(x^l_i), x^l_{i+1})\Big)=0, \end{aligned}\] and
\[\begin{aligned} \lim_{n\to\infty}\frac{1}{n}\sum_{i=-n}^{-1} d(f(y_i), y_{i+1}) < \lim_{n\to\infty}\frac{1}{n}\Big( \sum_{i=-n}^{-1}d(f(x^l_i), x^l_{i+1})+ {\rm diam}X+\epsilon \Big)=0. \end{aligned}\]
This means that \(\zeta=\{y_i:i\in\mathbb{Z}\}\) is an asymptotic average pseudo orbit of \(f\) through \(p.\) Since \(p\in AEVSh(f)\), there is \(z\in X\) such that \[\lim_{n\to\infty} \frac{1}{n}\sum_{i=0}^{n-1}d(f^i(z), y_i)=0, \ \mbox{and}\ \lim_{n\to\infty} \frac{1}{n}\sum_{i=-n}^{-1}d(f^i(z), y_i)=0.\]
Since \(y_i=x^l_i\) for \(i\not=0\) and \(y_0=p\), we also see that \[\lim_{n\to\infty} \frac{1}{n}\sum_{i=0}^{n-1}d(f^i(z), x^l_i)=0, \ \mbox{and}\\lim_{n\to\infty} \frac{1}{n}\sum_{i=-n}^{-1}d(f^i(z), x^l_i)=0.\]
This is a contradiction. ◻
Theorem 2.4. Let \(f\) be a homeomorphism of \(X\). \(AEVSh(f)=X\) if and only if \(f\) has the asymptotic average shadowing property.
Proof. It is clear that if \(f\) has the asymptotic average shadowing property then \(AEVSh(f)=X\). Thus we will prove the converse part. As Lemma 2.3, we have that if a homeomorphism \(f\) has the asymptotic average shadowing property then \(AEVSh(f)=X\). This completes the proof. ◻
Kawaguchi [4] proved that for any ergodic probability measure \(\mu\) of \(X\), if \(\mu(Sh(f))=1\) then \(f\) has the shadowing property, where \(\mu\) of \(X\) is ergodic if either \(\mu(A)=0\) or \(\mu(A)=1\) for any Borel set \(A\subset X\). For any Borel probability measure \(\mu\) of \(X\), we say that \(\mu\) is invariant for \(f\) if \(\mu(A)=\mu(f^{-1}(A))=\mu(f(A))\) for any Borel set \(A \subset X.\) Let \(\mathcal{M}_f(X)\) be the set of all invariant Borel probability measures of \(X\).
About the result, we introduced the following notion. We say that \(\mu\in\mathcal{M}_f(X)\) has the asymptotic average shadowing property if there is a Borel set \(A\subset X\) such that \(\mu(A)=1\) and any asymptotic average pseudo orbit \(\{x_i:i\in\mathbb{Z}\}\) through \(A(x_0\in A)\) can be asymptotic shadowed by a point in \(X\). It is observed that \(\mu\) has the asymptotic average shadowing property for \(f\), then \(\mu(AEVSh(f))=1.\)
Lemma 2.5. For any \(\mu\in\mathcal{M}_f(X)\), if \(\mu\) has the asymptotic average shadowing property then \(AEVSh(f)=X\).
Proof. For any \(x\in X\), there is an invariant measure \(\mu\) of \(X\) such that \(\mu(\omega(x))=1,\) where \(\omega(x)\) is the omega limit set of \(x.\) Since \(\mu\) has the asymptotic average shadowing property, \(\mu(AEVSh(f))=1.\) Then we can choose \(y\in \omega(x)\cap AEVSh(f).\) By Lemma 2.1, there is \(r>0\) such that \(B(y, r)\subset AEVSh(f).\) Since \(y\in\omega(f)\), there is \(j>0\) such that \(f^j(x)\in B(y, r)\subset AEVSh(f).\) By Lemma 2.2, \(AEVSh(f)\) is invariant. We have \(x\in AEVSh(f)\), and so \(AEVSh(f)=X.\) ◻
Theorem 2.6. Let \(f\) be a homeomorphism of \(X\). Any Borel probability measure \(\mu\) of \(X\) has the asymptotic average shadowing property if and only if \(f\) has the asymptotic average shadowing property.
Proof. Suppose that any Borel probability measure \(\mu\) of \(X\) has the asymptotic average shadowing property. By Lemma 2.5, we see that \(AEVSh(f)=X\). By Theorem 2.4, \(f\) has the asymptotic average shadowing property.
Conversely, , we assume that \(f\) has the asymptotic average shadowing property. By the definition of the asymptotic average shadowing property, it is clear that every Borel probability measure \(\mu\) of \(X\) has the asymptotic average shadowing property. ◻