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 μ of X has the asymptotic average shadowing property.
The pseudo-orbit tracing properties are very useful notion to investigate of the stability theory. Sakai [13] and Robinson [12] proved that a diffeomorphism \(f\) of a compact smooth manifold \(M\) has the robustly pseudo-orbit tracing property if and only if it is structurally stable. Later, a type of the pseudo-orbit tracing property was introduced in [1] which is called the average pseudo-orbit tracing property. The average pseudo-orbit tracing property has been studied by many people (see. [4, 6, 5, 7, 9, 10]). For example, Sakai proved in [14] that a diffeomorphism \(f\) of a two dimensional manifold \(M\) has the robustly average pseudo-orbit tracing property then it is Anosov. However, it is still open if the dimension of \(M\) is greater than 3. So we consider a weakly hyperbolic (partially hyperbolic) dynamical system. Regarding this system, Bonatti, Díaz and Turcat [2] proved that if a diffeomorphism \(f\) of the three dimensional manifold \(M\) is partially hyperbolic then it does not have the pseudo-orbit tracing property. Lee and Ahn [8] proved that if a diffeomorphism \(f\) of any dimensional manifold \(M\) is partially hyperbolic then it does not have the pseudo-orbit tracing property which is a generalization of the result of [2]. For the previous results of the pseudo-orbit tracing property([2, 8]), we deal with the average pseudo-orbit tracing property.
In this paper, we assume that \(M\) is a compact smooth Riemannian manifold. We denote \({\rm Diff}(M)\) as the set of all \(C^1\)-diffeomorphisms of \(M\) with the \(C^1\)-topology. Let \(d\) be a metric on \(M\) induced by a Riemannian metric \(\|\cdot\|\) on the tangent bundle \(TM.\)
For any \(f\in{\rm Diff}(M),\) we set \(Orb(x)=\{f^n(x): n\in\mathbb{Z}\}\) and it is said to be the orbit of \(x\). For any \(\delta>0\), a sequence \(\{x_i:i\in\mathbb{Z}\}\) is a \(\delta\) pseudo-orbit of \(f\) if for all \(i\in\mathbb{Z},\) we have \[d(f(x_i), x_{i+1})<\delta.\]
A diffeomorphism \(f:M\to M\), \(f\) has the pseudo-orbit tracing property if for any \(\epsilon>0\), there is \(\delta>0\) such that for any \(\delta\) pseudo-orbit \(\{x_i : i\in\mathbb{Z}\}\) of \(f\) can be \(\epsilon\) pseudo-orbit traced by some point of \(f\), that is, \[d(f^i(z), x_i)<\epsilon, \forall i\in\mathbb{Z}.\]
For any \(\delta>0\), a sequence \(\{x_i:i\in\mathbb{Z}\}\) is a \(\delta\) average pseudo-orbit of \(f\) if for any \(i\in\mathbb{Z}\), there is a positive integer \(N\) such that for any \(n \geq N\) and \(i\in\mathbb{Z}\), we have \[\frac{1}{n} \sum_{k=1}^n d(f(x_{k+i}), x_{k+i+1})<\delta.\]
Note that we can see that if a sequence \(\{x_i:i\in\mathbb{Z}\}\) is a \(\delta\) pseudo-orbit of \(f\) then the sequence is a \(\delta\) average pseudo-orbit of \(f.\)
For any \(\epsilon>0\), a \(\delta\) average pseudo-orbit \(\{x_i:i\in\mathbb{Z}\}\subset\Lambda\) is positively \(\epsilon\)-traced in average by some point \(z\in \Lambda\) if \[\limsup_{n\to\infty} \frac{1}{n} \sum_{i=0}^{n-1} d(f^i(z), x_i)<\epsilon.\] Analogously, we define negatively average pseudo-orbit tracing property.
Definition 2.1. Let \(\Lambda\) be a closed \(f\)-invariant set. We say that a diffeomorphism \(f\) of \(M\) has the average pseudo-orbit tracing property in \(\Lambda\) if for any \(\epsilon>0\), there is \(\delta>0\) such that any \(\delta\) average pseudo-orbit \(\{x_i:i\in\mathbb{Z}\}\subset\Lambda\) can be positively and negatively \(\epsilon\)-traced in average by some point \(z\in\Lambda\) (\(z\) is called an average pseudo-orbit tracing point ), that is, \[\limsup_{n\to\infty} \frac{1}{2n} \sum_{i=-n}^{n-1} d(f^i(z), x_i)<\epsilon.\]
Moreover, if \(\Lambda=M\) then we say that \(f\) has the average pseudo-orbit tracing property. Note that if the average pseudo-orbit tracing point contained in \(M\), then \(f\) has the average pseudo-orbit tracing property on \(\Lambda.\) That is, any \(\epsilon>0\), a \(\delta\) average pseudo-orbit \(\{x_i:i\in\mathbb{Z}\}\subset\Lambda\) is \(\epsilon\)-traced in average by some point \(z\in M\) if \[\limsup_{n\to\pm\infty} \frac{1}{2n} \sum_{i=-n}^{n-1} d(f^i(z), x_i)<\epsilon.\]
For a closed \(f\)-invariant set \(\Lambda\subset M\), \(\Lambda\) is called hyperbolic if if the tangent bundle \(T_{\Lambda}M\) has a continuous \(Df\)-invariant splitting \(E^s\oplus E^u\) and there exist constants \(C >0\) and \(\lambda\in(0, 1)\) such that \[\|Df^n|_{E^s(x)}\|\leq C\lambda^n \ \mbox{ and} \ \ \|Df^{-n}|_{E^u(x)}\|\leq C\lambda^n,\] for all \(x\in\Lambda\) and \(n\geq0.\) Moreover, if \(\Lambda=M\) then we say that \(f\) is Anosov.
A closed \(f\)-invariant set \(\Lambda\) has a dominated splitting if the tangent bundle \(T_{\Lambda}M\) has a continuous \(Df\)-invariant splitting \(E \oplus F\) and there exist constants \(C > 0\) and \(\lambda\in (0, 1)\) such that \[\|D_xf^n|_{E(x)}\|\cdot\|D_xf^{-n}|_{F(f^n(x))}\|\leq C\lambda^n,\] for all \(x \in\Lambda\) and \(n\geq0.\) It is clear that if a closed \(f\)-invariant set \(\Lambda\) is hyperbolic then \(\Lambda\) has a dominated splitting.
If either \(E\) is uniformly contracting or \(F\) is uniformly expanding then \(\Lambda\) is called partially hyperbolic for \(f\). It is clear that if \(\Lambda\) is hyperbolic then it has a dominated splitting, also, it is partially hyperbolic. If \(\Lambda=M\) then we say that \(f\) is a partially hyperbolic. A point \(x\in M\) is said to be periodic if there is \(n\in\mathbb{N}\) such that \(f^n(x)=x.\) Denote the set of all periodic points of \(f\) as \(P(f)\).
In [14], Sakai showed that the relation with the average pseudo-orbit tracing property on the two-dimensional manifold \(M\) and hyperbolicity, and for any dimensional manifold \(M\), Lee and Wen [10] proved that if a diffeomorphism \(f\) has the robustly average pseudo-orbit tracing property then it has a dominated splitting.
For a hyperbolic periodic point \(p\) of \(f\) with period \(\pi(p)\), the stable manifold of \(p\), \(W^s(p)=\{x\in M : d(f^{\pi(p)i}(x), p)\to 0\) as \(i\to\infty\}\) and the unstable manifold of \(p\), \(W^u(p)=\{x\in M: d(f^{\pi(p)i}(x), p)\to0\) as \(i\to-\infty\}.\) Let \(H(p)\) be the set of closures of the non-empty transverse intersection of \(W^s(p)\) and \(W^u(p).\)
In [6], Lee proved that if the homoclinic class \(H(p)\) has a dominated splitting, every periodic point in \(H(p)\) is hyperbolic, and a diffeomorphism \(f\) has the average pseudo-orbit tracing property on \(H(p),\) then \(H(p)\) is hyperbolic.
A subset \(\mathcal{R}\) is residual if it contains a dense \(G_{\delta}\) subset of \({\rm Diff}(M).\) A dynamic property of diffeomorphisms is called \(C^1\) generic if it holds on a residual set in \({\rm Diff}(M).\)
Lee proved in [7] that there is a residual subset \(\mathcal{R}\) of the set of two dimensional manifold \(M\) such that a diffeomorphism \(f\in\mathcal{R}\) has the average pseudo-orbit tracing property then it is Anosov. It is a generalization of result of Sakai [14]. Lee and Park proved in [9] that there is a residual subset \(\mathcal{R}\) of \({\rm Diff}(M)\) such that if a diffeomorpshim \(f\in\mathcal{R}\) has the average pseudo-orbit tracing property and every periodic points \(p\) and \(q\) are homoclinically related then it is Anosov, where the periodic points \(p\) and \(q\) are homoclinically related if \(W^s(p)\) and \(W^u(q)\) are non-empty transversal intersections and \(W^u(p)\) and \(W^s(q)\) are too.
In the paper, we consider the relationship with a type of the average pseudo-orbit tracing property and a homoclinic tangent associated with a hyperbolic periodic point \(p\).
Theorem 2.2(Theorem A). For any \(C^1\) generic \(f\in {\rm Diff}(M)\), if \(f\) have the average pseudo-orbit tracing property then \(f\) is partially hyperbolic.
Let \(M\) be as before, and let \(f:M\to M\) be a diffeomorphism.
Lemma 3.1. Let \(\Lambda\subset M\) be a closed \(f\)-invariant set. If \(f\) has the average pseudo-orbit tracing property, then \(f\) has the average pseudo-orbit tracing property in \(\Lambda.\)
Proof. Since \(\Lambda\) is a closed set, a closed set \(A\subset\Lambda\), there is \(r>0\) such that \(B(A, r) \subset \Lambda,\) where \(B(A, r)\) is a \(r\)-neighborhood of \(A\). For \(\epsilon=r\), let \(0<\delta\leq \epsilon\) be the number of the average pseudo-orbit tracing property for \(f.\) Let \(\{x_i:i\in\mathbb{Z}\}\subset \Lambda\) be a \(\delta\) average pseudo-orbit of \(f\) such that \(B(\overline{\{x_i:i\in\mathbb{Z}\}}, \epsilon)\subset \Lambda\). Assume that the \(\delta\) average pseudo-orbit \(\{x_i:i\in\mathbb{Z}\}\subset \Lambda\) can be \(\epsilon\) traced in average by a point \(z\in M\) such that \(Orb_f(z)\cap\Lambda=\emptyset.\) Then we see that \(d(f^i(z), x_i)\geq\epsilon\) for all \(i\in\mathbb{Z}.\) This is a contradiction by the average pseudo-orbit tracing property for \(f\). Thus the \(\delta\) average pseudo-orbit \(\{x_i:i\in\mathbb{Z}\}\subset \Lambda\) can be \(\epsilon\) traced in average by a point \(z\in M\) such that \(Orb_f(z)\cap\Lambda\not=\emptyset.\) Indeed, since \(\Lambda\) is \(f\)-invariant set, if there is \(j\in\mathbb{Z}\) such that \(f^j(z)\in\Lambda\) then \(f^i(f^j(z))\in\Lambda\) for all \(i\in\mathbb{Z}.\) Thus, \(z\) is a average pseudo-orbit tracing point in \(\Lambda,\) and so \(f\) has the average pseudo-orbit tracing property in \(\Lambda.\) ◻
Note that if \(f\) has the average pseudo-orbit tracing property, then \(f^i\) has the average pseudo-orbit tracing property, for all \(i\in\mathbb{Z}\setminus\{0\}\) (see [11, Theorem 3.2]).
For a hyperbolic periodic point \(p\) of \(f\), we assume that \(x\in W^s(p)\cap W^u(p)\setminus\{p\}\) a non-transversal point of \(f\). Let \(\Gamma=Orb(x)\cup\{Orb(p)\}.\)
Lemma 3.2. If \(f\) have the average pseudo-orbit tracing property in \(\Gamma,\) then the average pseudo-orbit tracing point \(z\) belongs to neither \(W^s(p)\) nor \(W^u(p)\).
Proof. For simplicity, assume that \(p\) is a fixed point, i.e., \(f(p) = p\). Since \(x \in W^s(p) \cap W^u(p) \setminus \{p\}\), for any sufficiently small \(r > 0\), there exists an integer \(l > 0\) such that \(f^l(x)\in B_{r/4}(p)\), \(f^{l-1}(x)\notin B_{r/4}(p)\), \(f^{-l}(x)\in B_{r/4}(p)\) and \(f^{-l+1}(x)\notin B_{r/4}(p).\) Then we have a sequence \(\xi_1=\{x_i:i=0, 1, \ldots, 2l+1\}\) such that \[\begin{aligned} & \{x_0=p, x_1=f^{-l}(x), x_2=f^{-l+1}(x), \ldots, \\& x_{l}=f^{-1}(x), x_{l+1}=x, x_{l+2}=f(x), \ldots, \\& x_{2l}=f^{l-1}(x), x_{2l+1}=p\}. \end{aligned}\]
The sequence \(\xi_1=\{x_i:i=0, 1, \ldots, 2l+1\}\) constitutes an \((r/4l)\)-average pseudo-orbit of \(f\). Indeed, we have: \[\begin{aligned} \frac{1}{2l+1}\sum_{i=0}^{2l} d(f(x_i), x_{i+1})&=\frac{1}{2l+1}(d(f(x_0), x_1)+d(f(x_{2l}), x_{2l+1}))\\&=\frac{1}{2l+1}(d(p, x_1)+d(f(f^{l-1}(x)), p))\\&<\frac{1}{2l+1}(\frac{r}{4}+\frac{r}{4})<\frac{r}{4l}. \end{aligned}\]
Next, we construct a sequence \(\xi_2=\{x_i:i=0, 1, \ldots, 4l+1\}\) by extending \(\xi_1\) with a sequence of \(p\):
\[\begin{aligned} &\{p(=x_0), f^{-l}(x), \ldots, f^{-1}(x), x, f(x), \ldots, f^{l-1}(x)(=x_{2l}), \\&\underbrace{p(=x_{2l+1}), p, \ldots, p, p(=x_{4l})}_{2l}, f^{-l}(x)(=x_{4l+1})\}. \end{aligned}\]
Then we see that \[\begin{aligned} \frac{1}{4l+1} \sum_{i=0}^{4l} d(f(x_i), x_{i+1})<\frac{1}{4l+1} \cdot \frac{3r}{4}<\frac{r}{4l}. \end{aligned}\]
Thus, \(\xi_2\) is also an \((r/4l)\)-average pseudo-orbit. By concatenating these sequences, we obtain an infinite sequence \(\xi_3\) such that for any \(k \in \mathbb{N}\): \[\begin{aligned} \frac{1}{2kl+1} \sum_{i=0}^{2kl} d(f(x_i), x_{i+1}) < \frac{1}{2kl+1} \cdot \frac{(2k+1)r}{4} < \frac{r}{4l}. \end{aligned}\]
Let \(\xi = \{x_i:i \in \mathbb{Z}\}\) be a bi-infinite sequence defined by \(\xi_4 = \{x_i = p : i \leq 0\}\) and \(\xi = \xi_4 \cup \xi_3\). One can easily verify that \(\xi\) is an \((r/4l)\)-average pseudo-orbit of \(f\).
Take \(\epsilon\in(0, r/4)\) and let \(0<\delta<\epsilon\) be the constant of the average pseudo-orbit tracing property such that \(\delta=r/4l.\)
We assume that the average pseudo-orbit tracing point \(z\in W^s(p)\) (the case \(z\in W^u(p)\) is similar). Since \(z\in W^s(p)\), there is \(j\geq l\) such that \(f^j(z)\in B_{r/4}(p)\cap W^s(p)\) and \(f^{j-1}(z)\notin B_{r/4}(p)\cap W^s(p).\) Then we see that \(f^{j+i}(z)\in B_{r/4}(p)\cap W^s(p)\) for all \(i\geq0,\) and \(d(f^j(z), f^{-l}(x))> r/4.\)
Since \(f\) has the average pseudo-orbit tracing property in \(\Gamma\), the \(\delta\) average pseudo-orbit \(\{x_i:i\in\mathbb{Z}\} \subset \Gamma\) can be negatively and positively \(\epsilon\) pseudo-orbit traced in average by the point \(z\in W^s(p)\).
By contradiction, suppose that \(f\) has the average pseudo-orbit tracing property in \(\Gamma\). Then there exists a point \(z \in \Gamma\) that \(\epsilon\)-traces \(\xi\) in average.
It is enough to show that the \(\delta\) average pseudo-orbit \(\{x_i:i\in\mathbb{Z}\} \subset \Gamma\) is positively \(\epsilon\) pseudo-orbit traced in average by the point \(z,\) that is, \[\limsup_{n\to\infty}\frac{1}{n} \sum_{i=0}^{n-1}d(f^i(z), x_i)<\epsilon.\]
For any \(k\in\mathbb{N}\) and \(n\in\mathbb{N}\), consider \(2kl\leq n\leq2(k+1)l\). We see that if \(n\to\infty\) then \(k\to\infty.\) Since \(d(f^{j+s}(z), f^{-l+i}(x))>r/4\) for all \(s\geq0\) and \(i=1, \ldots, 2l,\) we have that \[\begin{aligned} \frac{1}{n+1}\sum_{i=0}^n d(f^i(z), x_i)&\geq \frac{1}{2(k+1)l+1}\sum_{i=0}^{2kl} d(f^i(z), x_i) \\&>\frac{1}{2(k+1)l+1} \cdot \frac{r}{4}\cdot 2(k-2)l=\frac{(k-2)lr}{4((k+1)l+1)}. \end{aligned}\]
Since \(n\to\infty\) then \(k\to\infty\), we see that \[\begin{aligned} &\lim_{n\to\infty}\frac{1}{n+1}\sum_{i=0}^n d(f^i(z), x_i)>\lim_{k\to\infty} \frac{(k-2)lr}{4((k+1)l+1)} =\frac{r}{4} >\epsilon. \end{aligned}\]
This contradicts the assumption that \(z\) \(\epsilon\)-traces \(\xi\) in average. Therefore, \(f\) does not have the average pseudo-orbit tracing property in \(\Gamma = \text{Orb}(x) \cup \{Orb(p)\}\). ◻
Lemma 3.3. If \(f\) have the average pseudo-orbit tracing property in \(\Gamma,\) then the average pseudo-orbit tracing point \(z\) not belong to \(Orb(p)\).
Proof. Take \(\epsilon\in(0, r/4)\) and let \(0<\delta<\epsilon\) be the constant of the average pseudo-orbit tracing property such that \(\delta=r/4l\). Then as in the proof of Lemma 3.2, we make a \(\delta\) average pseudo-orbit \[\begin{aligned} \{x_i:i\in\mathbb{Z}\}=\{ \ldots, p, f^{-l}(x), \ldots, f^{-1}(x), x, f(x), \ldots, f^{l-1}(x), p, \ldots, p, f^{-l}(x), \ldots \} \subset \Gamma. \end{aligned}\]
By contradiction, suppose that the \(\delta\) average pseudo-orbit \(\{x_i:i\in\mathbb{Z}\} \subset \Gamma\) is positively \(\epsilon\) pseudo-orbit traced in average by the point \(z=p,\) that is, \[\limsup_{n\to\infty}\frac{1}{n} \sum_{i=0}^{n-1}d(f^i(z), x_i)<\epsilon.\]
Since \(d(p, f^i(x))>r/4\) for all \(i=-l, -l+1, \ldots, l-2,\) we see that for any \(k\in\mathbb{N}\) and \(n\in \mathbb{N}\), \(2kl\leq n\leq 2(k+1)l\). For the sequence \(\xi_1\) in Lemma 3.2, we see that the number of elements of the set \(\{i: d(p, x_i)>r/4, i=1, \ldots, 2l\}\) is \(2l\). Then as in the proof of Lemma 3.2, we know that \[\begin{aligned} &\lim_{n\to\infty}\frac{1}{n+1}\sum_{i=0}^n d(f^i(z), x_i)>\lim_{k\to\infty} \frac{(k-2)lr}{4((k+1)l+1)} =\frac{r}{4} >\epsilon. \end{aligned}\]
This means that the \(\delta\) pseudo-orbit \(\{x_i:i\in\mathbb{Z}\}\) is not positively \(\epsilon\) pseudo-orbit traced in average by the \(z=p\), whis is a contradiction ◻
Lemma 3.4. Let \(\epsilon>0\) and \(\delta>0\) be as in definition 2.1. Then \(f\) does not satisfy the average pseudo-orbit tracing property in \(\Gamma.\)
Proof. By contradiction, suppose that \(f\) has the average pseudo-orbit tracing property in \(\Gamma=Orb(x)\cup\{Orb(p)\}.\) As in the proof of Lemma 3.2 and Lemma 3.3, we can make a \(\delta\) average pseudo-orbit \(\{x_i:i\in\mathbb{Z}\}\subset \Gamma\). Since \(f\) has the average pseudo-orbit tracing property in \(\Gamma\), we can find the point \(z\in Orb(x)\) or \(z=p\) such that \[\limsup_{n\to\infty}\frac{1}{2n}\sum_{i=-n}^{n-1}d(f^i(z), x_i)<\epsilon.\] However, it is a contradiction by Lemma 3.2 and Lemma 3.3. ◻
Proposition 3.5. Let \(\epsilon>0\) and \(\delta>0\) be as in Definition 2.1. If \(f\) has the average pseudo-orbit tracing property and a \(\delta\) average pseudo-orbit \(\{x_i:i\in\mathbb{Z}\}\subset\Gamma\) as in Lemma 3.2, then an average pseudo-orbit tracing point is in \(\Gamma.\)
Proof. As in Lemma 3.2, we consider the sequence \(\{x_i:i\in\mathbb{Z}\}\subset\Gamma\) which is a \(\delta\) average pseudo-orbit of \(f\). Since \(f\) has the average pseudo-orbit tracing property, there is \(z\in M\) such that \[\limsup_{n\to\pm\infty}\frac{1}{2n}\sum_{i=-n}^{n-1}d(f^i(z), x_i)<\epsilon.\]
Assume that the point \(z\in M\setminus \Gamma\).
Case 1. Consider that if \(z\in M\setminus W^s(p)\cup W^u(p)\cup Orb(p)\) then \(f^i(z)\) can not converges to \(p\) as \(i\pm\infty\) and by \(\lambda\)-lemma \(f^i(W^s(p))\to W^s(p)\) as \(i\to-\infty\) and \(f^i(W^u(p))\to W^u(p)\) as \(i\to\infty,\) and thus there is \(j\in\mathbb{Z}\) such that \(d(f^{j+k}(z), x_{j+k})\geq\epsilon\) for all \(k\in\mathbb{Z}.\) Then we see that the point \(z\) is not average pseudo-orbit tracing point of \(f\).
Case 2. Consider that if \(z\in W^s(p)\cup W^u(p)\cup Orb(p)\) then by Lemma 3.2 and Lemma 3.3 the \(\delta\) average pseudo-orbit \(\{x_i:i\in\mathbb{Z}\}\subset\Gamma\) can not be \(\epsilon\) pseudo-orbit traced in average by \(z\).
By Case 1 and Case 2, the \(\delta\) average pseudo-orbit \(\{x_i:i\in\mathbb{Z}\}\subset \Gamma\) can be \(\epsilon\) pseudo-orbit traced in average by \(z\in\Gamma\). ◻
We say that \(f\) has a homoclinic tangency associated with a periodic point \(p\) if \(W^s(p)\) and \(W^u(p)\) are non-transversal intersections. Denote by \(\mathcal{HT}\) the set of all diffeomorphisms having the homoclinic tangency associated with periodic points. The following was proved in [3].
Lemma 3.6. There is a residual subset \(\mathcal{G}\) in \({\rm Diff}(M)\) such that for given \(f\in\mathcal{G}\), either \(f\) has a homoclinic tangency associated with a periodic point \(p\) or \(f\) is partially hyperbolic.
Proof of Theorem 2.2. Let \(f\in\mathcal{G}\subset{\rm Diff}(M)\) and \(f\) have the average pseudo-orbit tracing property. For simplicity, we assume that \(f(p)=p.\) For any point \(x\in W^s(p)\cap W^u(p)\setminus\{p\}\), by Proposition 3.5 we assume that \(f\) the average pseudo-orbit tracing property in \(\Gamma.\) According to Lemma 3.3 and Lemma 3.4, \(f\) does not satisfy the average pseudo-orbit tracing property in \(\Gamma.\) Thus if a diffeomorphism \(f\) has the average pseudo-orbit tracing property in \(\Gamma\), then we have that \(f\notin\mathcal{HT}.\) By Lemma 3.6, if \(f\in\mathcal{G}\) has the average pseudo-orbit tracing property in \(\Gamma\), then \(f\) is partially hyperbolic. ◻