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The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property

The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property 1IntroductionThe tracking property has an important application in topological dynamical systems. In recent years, more and more scholars pay attention to it, and the relevant research results are shown in [1,2,3, 4,5,6, 7,8,9, 10,11,12, 13,14,15, 16,17]. Liang and Li [1] proved that the self-map ffhas the tracking property if and only if the shift map σ\sigma has the tracking property in the inverse limit space. Ji et al. [2] proved that the shift map has the Lipschitz shadowing property if and only if the self-map has the Lipschitz shadowing property in the inverse limit space. Wang and Zeng [3] gave the relationship between average tracking property and q̲\underline{q}-average tracking property. Wu [4] proved that the self-map ffhas the d¯\overline{d}-tracking property if and only if the shift map σ\sigma has the d¯\overline{d}-tracking property in the inverse limit space.The map ffhas GG-asymptotic tracking property if for each ε>0\varepsilon \gt 0there exists δ>0\delta \gt 0such that for any (G,δ)\left(G,\delta )-pseudo orbit {xi}i≥0{\left\{{x}_{i}\right\}}_{i\ge 0}of ff, and there exists a point y∈Yy\in Yand l≥0l\ge 0such that the sequence {xi}i=l∞{\left\{{x}_{i}\right\}}_{i=l}^{\infty }is (G,ε)\left(G,\varepsilon )shadowed by point yy. We obtained the equivalent condition of the GG-asymptotic tracking property in metric GG-space.The map ffhas GG-Lipschitz tracking property if there exists positive constant LLand δ0{\delta }_{0}such that for any 0<δ<δ00\lt \delta \lt {\delta }_{0}and any (G,δ)\left(G,\delta )-pseudo orbit {xi}i≥0{\left\{{x}_{i}\right\}}_{i\ge 0}of ff, there exists a point x∈Xx\in Xsuch that the sequence {xi}i≥0{\left\{{x}_{i}\right\}}_{i\ge 0}is (G,Lδ)\left(G,L\delta )shadowed by point xx(see [18]). We proved that the map ffhas the GG-Lipschitz tracking property if and only if the shift map σ\sigma has the G¯\overline{G}-Lipschitz tracking property. The main results are as follows in this paper.Theorem 1.1Let (X,d)\left(X,d)be a compact metric G-space, the map f:X→Xf:X\to Xbe an equivalent map and the metric d be invariant to the topological group G, where G is exchangeable. Then, the map f has the G-asymptotic tracking property if and only if for any ε>0\varepsilon \gt 0, there exists 0<δ<ε0\lt \delta \lt \varepsilon and l≥0l\ge 0such that if {xk}k=0∞{\left\{{x}_{k}\right\}}_{k=0}^{\infty }is (G,δ)\left(G,\delta )-pseudo orbit of the map f, then there exists a point y in X such that {fl(xk)}k=0∞{\{{f}^{l}\left({x}_{k})\}}_{k=0}^{\infty }is (G,ε)\left(G,\varepsilon )shadowed by point y.Theorem 1.2Let (X,d)\left(X,d)be a compact metric G-space, (Xf,G¯,d¯,σ)\left({X}_{f},\overline{G},\overline{d},\sigma )be the inverse limit space of (X,G,d,f)\left(X,G,d,f)and the map f:X⟶Xf:X\hspace{0.33em}\longrightarrow \hspace{0.33em}Xbe an equivalent surjection. If the map f is an Lipschitz map with Lipschitz constant L, then we have that the map f has the G-Lipschitz tracking property if and only if the shift map σ\sigma has the G¯\overline{G}-Lipschitz tracking property.2The equivalent condition of GG-asymptotic tracking propertyIn this section, we present some concepts that may be used in the following. The concept of metric GG-space and equivariant map can be found in [17].Definition 2.1[19] Let (X,d)\left(X,d)be a metric space and ffbe a continuous map from XXto XX. The map ffis called to be uniformly continuous if for any ε>0\varepsilon \gt 0there exists 0<δ<ε0\lt \delta \lt \varepsilon such that d(x,y)<δd\left(x,y)\lt \delta implies d(f(x),f(y))<εd(f\left(x),f(y))\lt \varepsilon for all x,y∈Xx,y\in X.Definition 2.2[20] Let (X,d)\left(X,d)be a metric GG-space. The metric ddis said to be invariant to topological group GGprovided that d(gx,gy)=d(x,y)d\left(gx,gy)=d\left(x,y)for all x,y∈Xx,y\in Xand g∈Gg\in G.Definition 2.3Let (X,dX,d) be a metric GG-space and ffbe a continuous map from XXto XX. The map ffhas GG-asymptotic tracking property if for each ε>0\varepsilon \gt 0there exists δ>0\delta \gt 0such that for any (G,δG,\delta )-pseudo orbit {xi}i≥0{\left\{{x}_{i}\right\}}_{i\ge 0}of ff, there exists a point y∈Yy\in Yand l≥0l\ge 0such that the sequence {xi}i=l∞{\left\{{x}_{i}\right\}}_{i=l}^{\infty }is (G,ε)\left(G,\varepsilon )shadowed by point yy.Lemma 2.4Let (X,d)\left(X,d)be a compact metric G-space, the map f:X→Xf:X\to Xbe an equivalent map, the metric d be invariant to the topological group G, where G is exchangeable and m>0m\gt 0. Then, for any ε>0\varepsilon \gt 0, there exists 0<δ<ε0\lt \delta \lt \varepsilon such that if for any k≥0k\ge 0, there exists gk∈G{g}_{k}\in Gsuch that d(gkf(xk),xk+1)<δd\left({g}_{k}f\left({x}_{k}),{x}_{k+1})\lt \delta , then we have d(gk+m−1gk+m−2gk+m−3⋯gk+1gkfm(xk),xm+k)<εd\left({g}_{k+m-1}{g}_{k+m-2}{g}_{k+m-3}\cdots {g}_{k+1}{g}_{k}{f}^{m}\left({x}_{k}),{x}_{m+k})\lt \varepsilon .ProofBy continuity of the map ff, for any ε>0\varepsilon \gt 0and 0≤i<m0\le i\lt m, there exists 0<δ<ε0\lt \delta \lt \varepsilon such that d(x,y)<δd\left(x,y)\lt \delta implies (1)d(fi(x),fi(y))<εm.d({f}^{i}\left(x),{f}^{i}(y))\lt \frac{\varepsilon }{m}.Suppose that for any k>0k\gt 0, there exists gk∈G{g}_{k}\in Gsuch that d(gkf(xk),xk+1)<δ.d\left({g}_{k}f\left({x}_{k}),{x}_{k+1})\lt \delta .\hspace{0.16em}According to the equivalent definition of the map ffand (1), for any k>0k\gt 0and 0≤i<m0\le i\lt m, it follows that d(gkfi+1(xk),fi(xk+1))<εm.d\left({g}_{k}{f}^{i+1}\left({x}_{k}),{f}^{i}\left({x}_{k+1}))\lt \frac{\varepsilon }{m}.\hspace{0.65em}Then, d(gkfm(xk),fm−1(xk+1))<εm.d\left({g}_{k}{f}^{m}\left({x}_{k}),{f}^{m-1}\left({x}_{k+1}))\lt \frac{\varepsilon }{m}.d(gk+1fm−1(xk+1),fm−2(xk+2))<εm.d\left({g}_{k+1}{f}^{m-1}\left({x}_{k+1}),{f}^{m-2}\left({x}_{k+2}))\lt \frac{\varepsilon }{m}.d(gk+2fm−2(xk+2),fm−3(xk+3))<εm.d\left({g}_{k+2}{f}^{m-2}\left({x}_{k+2}),{f}^{m-3}\left({x}_{k+3}))\lt \frac{\varepsilon }{m}.\hspace{0.5em}⋯⋯\cdots \cdots d(gk+m−2f2(xk+m−2),f(xk+m−1))<εm.d\left({g}_{k+m-2}{f}^{2}\left({x}_{k+m-2}),f\left({x}_{k+m-1}))\lt \frac{\varepsilon }{m}.d(gk+m−1f(xk+m−1),xk+m)<εm.d\left({g}_{k+m-1}f\left({x}_{k+m-1}),{x}_{k+m})\lt \frac{\varepsilon }{m}.Since the metric ddis invariant to the topological group GGand GGis exchangeable, we have d(gkgk+1gk+2⋯gk+m−1fm(xk),gk+1gk+2⋯gk+m−1fm−1(xk+1))<εm.d\left({g}_{k}{g}_{k+1}{g}_{k+2}\cdots {g}_{k+m-1}{f}^{m}\left({x}_{k}),{g}_{k+1}{g}_{k+2}\cdots {g}_{k+m-1}{f}^{m-1}\left({x}_{k+1}))\lt \frac{\varepsilon }{m}.d(gk+1gk+2gk+3⋯gk+m−1fm−1(xk+1),gk+2gk+3⋯gk+m−1fm−2(xk+2))<εm.d\left({g}_{k+1}{g}_{k+2}{g}_{k+3}\cdots {g}_{k+m-1}{f}^{m-1}\left({x}_{k+1}),{g}_{k+2}{g}_{k+3}\cdots {g}_{k+m-1}{f}^{m-2}\left({x}_{k+2}))\lt \frac{\varepsilon }{m}.d(gk+2gk+3⋯gk+m−1fm−2(xk+2),gk+3⋯gk+m−1fm−3(xk+3))<εm.d\left({g}_{k+2}{g}_{k+3}\cdots {g}_{k+m-1}{f}^{m-2}\left({x}_{k+2}),{g}_{k+3}\cdots {g}_{k+m-1}{f}^{m-3}\left({x}_{k+3}))\lt \frac{\varepsilon }{m}.⋯⋯\cdots \cdots d(gk+m−2gk+m−1f2(xk+m−2),gk+m−1f(xk+m−1))<εm.d\left({g}_{k+m-2}{g}_{k+m-1}{f}^{2}\left({x}_{k+m-2}),{g}_{k+m-1}f\left({x}_{k+m-1}))\lt \frac{\varepsilon }{m}.d(gk+m−1f(xk+m−1),xk+m)<εm.d\left({g}_{k+m-1}f\left({x}_{k+m-1}),{x}_{k+m})\lt \frac{\varepsilon }{m}.Therefore, d(gkgk+1gk+2⋯gk+m−1fm(xk),xm+k)<d(gkgk+1gk+2⋯gk+m−1fm(xk),gk+1gk+2⋯gk+m−1fm−1(xk+1))+d(gk+1gk+2gk+3⋯gk+m−1fm−1(xk+1),gk+2gk+3⋯gk+m−1fm−2(xk+2))+d(gk+2gk+3⋯gk+m−1fm−2(xk+2),gk+3⋯gk+m−1fm−3(xk+3))+⋯⋯+d(gk+m−2gk+m−1f2(xk+m−2),gk+m−1f(xk+m−1))+d(gk+m−1f(xk+m−1),xk+m)<εm+εm+εm+⋯⋯+εm=ε.□\hspace{5.9em}\begin{array}{l}d\left({g}_{k}{g}_{k+1}{g}_{k+2}\cdots {g}_{k+m-1}{f}^{m}\left({x}_{k}),{x}_{m+k})\\ \hspace{1.0em}\lt d\left({g}_{k}{g}_{k+1}{g}_{k+2}\cdots {g}_{k+m-1}{f}^{m}\left({x}_{k}),{g}_{k+1}{g}_{k+2}\cdots {g}_{k+m-1}{f}^{m-1}\left({x}_{k+1}))\\ \hspace{2.0em}+d\left({g}_{k+1}{g}_{k+2}{g}_{k+3}\cdots {g}_{k+m-1}{f}^{m-1}\left({x}_{k+1}),{g}_{k+2}{g}_{k+3}\cdots {g}_{k+m-1}{f}^{m-2}\left({x}_{k+2}))\hspace{2.0em}+d\left({g}_{k+2}{g}_{k+3}\cdots {g}_{k+m-1}{f}^{m-2}\left({x}_{k+2}),{g}_{k+3}\cdots {g}_{k+m-1}{f}^{m-3}\left({x}_{k+3}))\\ \hspace{1.0em}\hspace{1.0em}+\cdots \cdots +d\left({g}_{k+m-2}{g}_{k+m-1}{f}^{2}\left({x}_{k+m-2}),{g}_{k+m-1}f\left({x}_{k+m-1}))+d\left({g}_{k+m-1}f\left({x}_{k+m-1}),{x}_{k+m})\\ \hspace{1.0em}\lt \frac{\varepsilon }{m}+\frac{\varepsilon }{m}+\frac{\varepsilon }{m}+\cdots \cdots +\frac{\varepsilon }{m}=\varepsilon .\end{array}\hspace{3.75em}<mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/"xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">\square </mml:mpadded>Theorem 2.5Let (X,d)\left(X,d)be a compact metric G-space, the map f:X→Xf:X\to Xbe an equivalent map and the metric d be invariant to the topological group G where G is exchangeable. Then, the map f has the G-asymptotic tracking property if and only if for any ε>0\varepsilon \gt 0there exists 0<δ<ε0\lt \delta \lt \varepsilon and l≥0l\ge 0such that if {xk}k=0∞{\left\{{x}_{k}\right\}}_{k=0}^{\infty }is (G,δ)\left(G,\delta )-pseudo orbit of the map f, then there exists a point y in X such that {fl(xk)}k=0∞{\{{f}^{l}\left({x}_{k})\}}_{k=0}^{\infty }is (G,ε)\left(G,\varepsilon )shadowed by point y.Proof(Necessity) Suppose that the map ffhas the GG-asymptotic tracking property. Then, for each ε>0\varepsilon \gt 0, there exists 0<τ<ε20\lt \tau \lt \frac{\varepsilon }{2}such that for any (G,τ)\left(G,\tau )-pseudo orbit {xk}k≥0{\left\{{x}_{k}\right\}}_{k\ge 0}of ff, there exists a point y∈Xy\in Xand l≥0l\ge 0such that the sequence {xk}k=l∞{\left\{{x}_{k}\right\}}_{k=l}^{\infty }is (G,ε2)\left(G,\frac{\varepsilon }{2})shadowed by point yy. If l=0l=0, the results are obvious. Now we assume l>0l\gt 0. Since the map ffis uniformly continuous, for given ε2l>0\frac{\varepsilon }{2l}\gt 0and any 0≤i<l0\le i\lt l, there exists 0<δ<minε2l,τ0\lt \delta \lt \min \left\{\frac{\varepsilon }{2l},\tau \right\}such that d(x,y)<δd\left(x,y)\lt \delta implies (2)d(fi(x),fi(y))<ε2l.d({f}^{i}\left(x),{f}^{i}(y))\lt \frac{\varepsilon }{2l}.Let {xk}k=0∞{\left\{{x}_{k}\right\}}_{k=0}^{\infty }be (G,δ)\left(G,\delta )-pseudo orbit of the map ff. Then, for any k≥0k\ge 0, there exists tk∈G{t}_{k}\in Gsuch that d(tkf(xk),xk+1)<δ.d\left({t}_{k}f\left({x}_{k}),{x}_{k+1})\lt \delta .\hspace{0.16em}By (2) and the equivalent definition of the map ff, for any k≥0k\ge 0and 0≤i<l0\le i\lt l, we have that (3)d(tkfi+1(xk),fi(xk+1))<ε2l.d\left({t}_{k}{f}^{i+1}\left({x}_{k}),{f}^{i}\left({x}_{k+1}))\lt \frac{\varepsilon }{2l}.Noting that the metric ddis invariant to the topological group GG, where GGis exchangeable and (3), we have d(tktk+1tk+2⋯tk+l−2tk+l−1fl(xk),tk+1tk+2⋯tk+l−2tk+l−1fl−1(xk+1))<ε2l,d\left({t}_{k}{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l}\left({x}_{k}),{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-1}\left({x}_{k+1}))\lt \frac{\varepsilon }{2l},d(tk+1tk+2tk+3⋯tk+l−2tk+l−1fl−1(xk+1),tk+2tk+3⋯tk+l−2tk+l−1fl−2(xk+2))<ε2l,d\left({t}_{k+1}{t}_{k+2}{t}_{k+3}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-1}\left({x}_{k+1}),{t}_{k+2}{t}_{k+3}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-2}\left({x}_{k+2}))\lt \frac{\varepsilon }{2l},d(tk+2tk+3⋯tk+l−2tk+l−1fl−2(xk+2),tk+3⋯tk+l−2tk+l−1fl−3(xk+3))<ε2l,d\left({t}_{k+2}{t}_{k+3}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-2}\left({x}_{k+2}),{t}_{k+3}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-3}\left({x}_{k+3}))\lt \frac{\varepsilon }{2l},⋯⋯\cdots \cdots d(tk+l−2tk+l−1f2(xk+l−2),tk+l−1f(xk+l−1))<ε2l,d\left({t}_{k+l-2}{t}_{k+l-1}{f}^{2}\left({x}_{k+l-2}),{t}_{k+l-1}f\left({x}_{k+l-1}))\lt \frac{\varepsilon }{2l},d(tk+l−1f(xk+l−1),xk+l)<ε2l,d\left({t}_{k+l-1}f\left({x}_{k+l-1}),{x}_{k+l})\lt \frac{\varepsilon }{2l},and thus, d(tktk+1tk+2⋯tk+l−2tk+l−1fl(xk),xk+l)<d(tktk+1tk+2⋯tk+l−2tk+l−1fl(xk),tk+1tk+2⋯tk+l−2tk+l−1fl−1(xk+1))+d(tk+1tk+2tk+3⋯tk+l−2tk+l−1fl−1(xk+1),tk+2tk+3⋯tk+l−2tk+l−1fl−2(xk+2))+d(tk+2tk+3⋯tk+l−2tk+l−1fl−2(xk+2),tk+3⋯tk+l−2tk+l−1fl−3(xk+3))+⋯⋯+d(tk+l−2tk+l−1f2(xk+l−2),tk+l−1f(xk+l−1))+d(tk+l−1f(xk+l−1),xk+l)<ε2l+ε2l+ε2l+⋯⋯+ε2l=ε2.\begin{array}{l}d\left({t}_{k}{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l}\left({x}_{k}),{x}_{k+l})\\ \hspace{1.0em}\lt d\left({t}_{k}{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l}\left({x}_{k}),{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-1}\left({x}_{k+1}))\\ \hspace{2.0em}+d\left({t}_{k+1}{t}_{k+2}{t}_{k+3}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-1}\left({x}_{k+1}),{t}_{k+2}{t}_{k+3}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-2}\left({x}_{k+2}))\hspace{2.0em}+d\left({t}_{k+2}{t}_{k+3}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-2}\left({x}_{k+2}),{t}_{k+3}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-3}\left({x}_{k+3}))\\ \hspace{2.0em}+\cdots \cdots +d\left({t}_{k+l-2}{t}_{k+l-1}{f}^{2}\left({x}_{k+l-2}),{t}_{k+l-1}f\left({x}_{k+l-1}))+d\left({t}_{k+l-1}f\left({x}_{k+l-1}),{x}_{k+l})\\ \hspace{1.0em}\lt \frac{\varepsilon }{2l}+\frac{\varepsilon }{2l}+\frac{\varepsilon }{2l}+\cdots \cdots +\frac{\varepsilon }{2l}=\frac{\varepsilon }{2}.\end{array}So for any k≥0k\ge 0, we have (4)d(tktk+1tk+2⋯tk+l−2tk+l−1fl(xk),xk+l)<ε2.d\left({t}_{k}{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l}\left({x}_{k}),{x}_{k+l})\lt \frac{\varepsilon }{2}.\hspace{1.1em}Since the map ffhas the GG-asymptotic tracking property, for any k≥0k\ge 0, there exists gk∈G{g}_{k}\in Gand y∈Yy\in Ysuch that d(fk(y),gkxk+l)<ε2.d({f}^{k}(y),{g}_{k}{x}_{k+l})\lt \frac{\varepsilon }{2}.\hspace{2.7em}Since the metric ddis invariant to the topological group GG, then (5)d(gk−1fk(y),xk+l)<ε2.d\left({g}_{k}^{-1}{f}^{k}(y),{x}_{k+l})\lt \frac{\varepsilon }{2}.\hspace{2.5em}By (4) and (5), for any k≥0k\ge 0, we obtain d(tktk+1tk+2⋯tk+l−2tk+l−1fl(xk),gk−1fk(y))<d(tktk+1tk+2⋯tk+l−2tk+l−1fl(xk),xk+l)+d(xk+l,gk−1fk(y))<ε.d\left({t}_{k}{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l}\left({x}_{k}),{g}_{k}^{-1}{f}^{k}(y))\lt d\left({t}_{k}{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l}\left({x}_{k}),{x}_{k+l})+d\left({x}_{k+l},{g}_{k}^{-1}{f}^{k}(y))\lt \varepsilon .Together with the fact that the metric ddis invariant to the topological group GGagain, it follows that d(fk(y),gktktk+1tk+2⋯tk+l−2tk+l−1fl(xk))<ε.d({f}^{k}(y),{g}_{k}{t}_{k}{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l}\left({x}_{k}))\lt \varepsilon .Hence, the sequence {fl(xk)}k=0∞{\{{f}^{l}\left({x}_{k})\}}_{k=0}^{\infty }is (G,ε)\left(G,\varepsilon )shadowed by point yy.(Sufficiency) Suppose that for any ε>0\varepsilon \gt 0there exists 0<τ<ε0\lt \tau \lt \varepsilon and l≥0l\ge 0such that if {xk}k=0∞{\left\{{x}_{k}\right\}}_{k=0}^{\infty }is (G,τ)\left(G,\tau )-pseudo orbit of the map ff, then there exists a point zzin XXsuch that {fl(xk)}k=0∞{\{{f}^{l}\left({x}_{k})\}}_{k=0}^{\infty }is (G,ε2)\left(G,\frac{\varepsilon }{2}\right)shadowed by point zz. If l=0l=0, the results are obvious. Now we assume l>0l\gt 0. Since the map ffis uniformly continuous, for given ε>0\varepsilon \gt 0and any 0≤i<l0\le i\lt l, there exists 0<δ0<τ0\lt {\delta }_{0}\lt \tau such that d(x,y)<δd\left(x,y)\lt \delta implies d(fi(x),fi(y))<ε2.d({f}^{i}\left(x),{f}^{i}(y))\lt \frac{\varepsilon }{2}.Let {xk}k=0∞{\{{x}_{k}\}}_{k=0}^{\infty }be (G,δ0)\left(G,{\delta }_{0})-pseudo orbit of the map ff. Then, there exists z∈Xz\in Xsuch that for any k≥0k\ge 0there exists pk∈G{p}_{k}\in Gsuch that (6)dfk(z),pkfl(xk))<ε2.d\left({f}^{k}\left(z)\left,{p}_{k}{f}^{l}\left({x}_{k})\right)\lt \frac{\varepsilon }{2}.\right.In addition, for any k≥0k\ge 0, there exists sk∈G{s}_{k}\in Gsuch that d(skf(xk),xk+1)<δ0.d\left({s}_{k}f\left({x}_{k}),{x}_{k+1})\lt {\delta }_{0}.\hspace{1em}By Lemma 2.9, for any k≥0k\ge 0, we have that (7)d(sksk+1sk+2⋯sk+l−2sk+l−1fl(xk),xk+l)<ε2.d\left({s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{f}^{l}\left({x}_{k}),{x}_{k+l})\lt \frac{\varepsilon }{2}.Since the metric ddis invariant to the topological group GGand equations (6) and (7), we have dsksk+1sk+2⋯sk+l−2sk+l−1fk(z),sksk+1sk+2⋯sk+l−2sk+l−1pkfl(xk))<ε2d\left({s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{f}^{k}\left(z),{s}_{k}{s}_{k+1}{s}_{k+2}\left\cdots {s}_{k+l-2}{s}_{k+l-1}{p}_{k}{f}^{l}\left({x}_{k})\right)\lt \frac{\varepsilon }{2}\right.and d(pksksk+1sk+2⋯sk+l−2sk+l−1f(xk),pkxk+l)<ε2.d\left({p}_{k}{s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}f\left({x}_{k}),{p}_{k}{x}_{k+l})\lt \frac{\varepsilon }{2}.Since GGis exchangeable, we have that d(sksk+1sk+2⋯sk+l−2sk+l−1fk(z),pkxk+l)<d(sksk+1sk+2⋯sk+l−2sk+l−1fk(z),sksk+1sk+2⋯sk+l−2sk+l−1pkfl(xk)+d(sksk+1sk+2⋯sk+l−2sk+l−1pkfl(xk),pkxk+l)=d(sksk+1sk+2⋯sk+l−2sk+l−1fk(z),sksk+1sk+2⋯sk+l−2sk+l−1pkfl(xk)+d(pksksk+1sk+2⋯sk+l−2sk+l−1fl(xk),pkxk+1)<ε2+ε2<ε.\begin{array}{l}d\left({s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{f}^{k}\left(z),{p}_{k}{x}_{k+l})\\ \hspace{1.0em}\lt d({s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{f}^{k}\left(z),{s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{p}_{k}{f}^{l}\left({x}_{k})+d\left({s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{p}_{k}{f}^{l}\left({x}_{k}),{p}_{k}{x}_{k+l})\\ \hspace{1.0em}=d({s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{f}^{k}\left(z),{s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{p}_{k}{f}^{l}\left({x}_{k})+d\left({p}_{k}{s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{f}^{l}\left({x}_{k}),{p}_{k}{x}_{k+1})\\ \hspace{1.0em}\lt \frac{\varepsilon }{2}+\frac{\varepsilon }{2}\lt \varepsilon .\end{array}\hspace{0.25em}Hence, the map ffhas the GG-asymptotic tracking property. Thus, we complete the proof.□3GG-Lipschitz tracking propertyThe concept of the inverse limit spaces in this section under group action can be found in [21].Definition 3.1[5] Let(X,d)\left(X,d)be a metric space and f:X→Xf:X\to Xbe a continuous map.The map ffis said to be an Lipschitz map if there exists a positive constant L>0L\gt 0such that d(f(x),f(y))≤Ld(x,y)d(f\left(x),f(y))\le Ld\left(x,y)for all x,y∈Xx,y\in X.Definition 3.2[18] Let(X,d)\left(X,d)be a metric GG-space and f:X→Xf:X\to Xbe a continuous map. The map ffhas GG-Lipschitz tracking property if there exists positive constant LLand δ0{\delta }_{0}such that for any 0<δ<δ00\lt \delta \lt {\delta }_{0}and (G,δ)\left(G,\delta )-pseudo orbit {xi}i≥0{\left\{{x}_{i}\right\}}_{i\ge 0}of ffthere exists a point zzin XXsuch that the sequence {xi}i≥0{\left\{{x}_{i}\right\}}_{i\ge 0}is (G,Lδ)\left(G,L\delta )shadowed by point zz.Now, we give the proof of Theorem 3.3.Theorem 3.3Let (X,d)\left(X,d)be a compact metric G-space, (Xf,G¯,d¯,σ)\left({X}_{f},\overline{G},\overline{d},\sigma )be the inverse limit space of (X,G,d,f)\left(X,G,d,f)and the map f:X⟶Xf:X\hspace{0.33em}\longrightarrow \hspace{0.33em}Xbe an equivalent surjection. If the map f is an Lipschitz map with Lipschitz constant L, then we have that the map f has the G-Lipschitz tracking property if and only if the shift map σ\sigma has the G¯\overline{G}-Lipschitz tracking property.Proof(Necessity) Suppose that the map ffhas the GG-Lipschitz tracking property. Then, there exists positive constant L1{L}_{1}and ε1{\varepsilon }_{1}such that for any 0<ε<ε10\lt \varepsilon \lt {\varepsilon }_{1}and (G,ε)\left(G,\varepsilon )-pseudo orbit {xi}i=0∞{\left\{{x}_{i}\right\}}_{i=0}^{\infty }of the map ff, there exists a point x∈Xx\in Xsuch that {xi}i=0∞{\left\{{x}_{i}\right\}}_{i=0}^{\infty }is (G,L1ε)\left(G,{L}_{1}\varepsilon )shadowed by xx. By the compactness of XX, let M=diam(X)M={\rm{diam}}\left(X). For given ε>0\varepsilon \gt 0, choose a positive constant m>0m\gt 0such that M2m<ε\frac{M}{{2}^{m}}\lt \varepsilon . Let L2=Lm+Lm−12+Lm−222+⋯+L2m−1+12m,{L}_{2}={L}_{m}+\frac{{L}_{m-1}}{2}+\frac{{L}_{m-2}}{{2}^{2}}+\cdots +\frac{L}{{2}^{m-1}}+\frac{1}{{2}^{m}},L3=L1L22m,{L}_{3}={L}_{1}{L}_{2}{2}^{m},ε2=ε12m.{\varepsilon }_{2}=\frac{{\varepsilon }_{1}}{{2}^{m}}.For any 0<η<ε20\lt \eta \lt {\varepsilon }_{2}, let {y¯k}k=0∞=(yk0,yk1,yk2⋯){\{{\overline{y}}_{k}\}}_{k=0}^{\infty }=({y}_{k}^{0},{y}_{k}^{1},{y}_{k}^{2}\cdots )be (G,η)\left(G,\eta )-pseudo orbit of the shift map σ\sigma in Xf{X}_{f}. It is obvious that for each k≥0k\ge 0, there exists g¯k=(gk,gk,gk⋯)∈G¯{\overline{g}}_{k}=\left({g}_{k},{g}_{k},{g}_{k}\cdots )\in \overline{G}such that d¯(g¯kσ(y¯k),y¯k+1)<η.\overline{d}\left({\overline{g}}_{k}\sigma ({\overline{y}}_{k}),{\overline{y}}_{k+1})\lt \eta .From the definition of the metric d¯\overline{d}, for every k≥0k\ge 0, it follows that d(gkf(ykm),yk+1m)<2mη<ε1.\hspace{5.75em}d\left({g}_{k}f({y}_{k}^{m}),{y}_{k+1}^{m})\lt {2}^{m}\eta \lt {\varepsilon }_{1}.\hspace{6.1em}Thus, {ykm}k=0∞{\{{y}_{k}^{m}\}}_{k=0}^{\infty }is (G,2mη)\left(G,{2}^{m}\eta )-pseudo orbit of the map ff. By the GG-Lipschitz tracking property of the map ffin XX, there exists a point yyin XXsuch that for every k≥0k\ge 0there exists tk∈G{t}_{k}\in Gsuch that (8)d(fk(y),tkykm)<L12mη.d({f}^{k}(y),{t}_{k}{y}_{k}^{m})\lt {L}_{1}{2}^{m}\eta .Since the map ffis onto, we write y¯=(fm(y),fm−1(y),⋯,f(y),y,⋯)∈Xf.\hspace{5.85em}\overline{y}=({f}^{m}(y),{f}^{m-1}(y),\cdots \hspace{0.33em},f(y),y,\cdots )\in {X}_{f}.\hspace{5.85em}t¯k=(tk,tk,tk⋯)∈G¯.{\overline{t}}_{k}=\left({t}_{k},{t}_{k},{t}_{k}\cdots )\in \overline{G}.\hspace{5.85em}By the definition of the equivalent map ff, for any k≥0k\ge 0, we have d¯(σk(y¯),t¯ky¯k)<∑i=0i=md(fk+m−i(y),tkyki)2i+∑i=m+1∞M2i<∑i=0i=md(fm−i(fk(y)),fm−i(tkykm))2i+M2m<∑i=0i=md(fm−i(fk(y)),fm−i(tkykm))2i+ε≤∑i=0i=md(fm−i(fk(y)),fm−i(tkykm))2i.\begin{array}{rcl}\overline{d}\left({\sigma }^{k}(\overline{y}),{\overline{t}}_{k}{\overline{y}}_{k})& \lt & \mathop{\displaystyle \sum }\limits_{i=0}^{i=m}\frac{d({f}^{k+m-i}(y),{t}_{k}{y}_{k}^{i})}{{2}^{i}}+\mathop{\displaystyle \sum }\limits_{i=m+1}^{\infty }\frac{M}{{2}^{i}}\\ & \lt & \mathop{\displaystyle \sum }\limits_{i=0}^{i=m}\frac{d({f}^{m-i}({f}^{k}(y)),{f}^{m-i}\left({t}_{k}{y}_{k}^{m}))}{{2}^{i}}+\frac{M}{{2}^{m}}\\ & \lt & \mathop{\displaystyle \sum }\limits_{i=0}^{i=m}\frac{d({f}^{m-i}({f}^{k}(y)),{f}^{m-i}\left({t}_{k}{y}_{k}^{m}))}{{2}^{i}}+\varepsilon \\ & \le & \mathop{\displaystyle \sum }\limits_{i=0}^{i=m}\frac{d({f}^{m-i}({f}^{k}(y)),{f}^{m-i}\left({t}_{k}{y}_{k}^{m}))}{{2}^{i}}.\end{array}According to (8) and the definition of Lipschitz map ff, we obtain ∑i=0i=md(fm−i(fk(y)),fm−i(tkykm))2i<∑i=0i=mLm−id(fk(y),tkykm)2i<∑i=0i=mLm−iL12mη2i=L1L22mη.\mathop{\sum }\limits_{i=0}^{i=m}\frac{d({f}^{m-i}({f}^{k}(y)),{f}^{m-i}\left({t}_{k}{y}_{k}^{m}))}{{2}^{i}}\lt \mathop{\sum }\limits_{i=0}^{i=m}\frac{{L}^{m-i}d({f}^{k}(y),{t}_{k}{y}_{k}^{m})}{{2}^{i}}\lt \mathop{\sum }\limits_{i=0}^{i=m}\frac{{L}^{m-i}{L}_{1}{2}^{m}\eta }{{2}^{i}}={L}_{1}{L}_{2}{2}^{m}\eta .Then, for any k≥0k\ge 0, it follows that d¯(σk(y¯),t¯ky¯k)<L3η.\overline{d}\left({\sigma }^{k}(\overline{y}),{\overline{t}}_{k}{\overline{y}}_{k})\lt {L}_{3}\eta .Therefore, the shift map σ\sigma has the GG-Lipschitz tracking property.(Sufficiency) Next, we suppose that the shift map σ\sigma has the G¯\overline{G}-Lipschitz tracking property. Then, there exists positive constant L4{L}_{4}and ε3{\varepsilon }_{3}such that for any 0<δ′<ε30\lt {\delta }^{^{\prime} }\lt {\varepsilon }_{3}and (G¯,δ′)\left(\overline{G},{\delta }^{^{\prime} })-pseudo orbit {z¯k}k=0∞{\left\{{\overline{z}}_{k}\right\}}_{k=0}^{\infty }of the shift map σ\sigma there exists a point z¯∈Xf\overline{z}\in {X}_{f}such that {z¯k}k=0∞{\left\{{\overline{z}}_{k}\right\}}_{k=0}^{\infty }is (G¯,L4δ′)\left(\overline{G},{L}_{4}{\delta }^{^{\prime} })shadowed by point z¯.\overline{z}.For given δ′>0{\delta }^{^{\prime} }\gt 0, choose n>0n\gt 0such that M2n<δ′\frac{M}{{2}^{n}}\lt {\delta }^{^{\prime} }. Then, we write L5=Ln+Ln−12Ln−222+⋯+L2n−1+12n,{L}_{5}={L}_{n}+\frac{{L}_{n-1}}{2}\frac{{L}_{n-2}}{{2}^{2}}+\cdots +\frac{L}{{2}^{n-1}}+\frac{1}{{2}^{n}},L6=2nL4L5,{L}_{6}={2}^{n}{L}_{4}{L}_{5},ε4=ε3L5.{\varepsilon }_{4}=\frac{{\varepsilon }_{3}}{{L}_{5}}.For any 0<δ<ε40\lt \delta \lt {\varepsilon }_{4}, suppose that {xk}k=0∞{\left\{{x}_{k}\right\}}_{k=0}^{\infty }is (G,δ)\left(G,\delta )-pseudo orbit of the map ff. It is obvious that for each k≥0k\ge 0, there exists a point pk∈G{p}_{k}\in Gsuch that (9)d(pkf(xk),xk+1)<δ.d\left({p}_{k}f\left({x}_{k}),{x}_{k+1})\lt \delta .\hspace{1.7em}Since the map ffis onto, we write x¯k=(fn(xk),fn−1(xk),⋯,f(xk),xk,⋯)∈Xf.{\overline{x}}_{k}=({f}^{n}\left({x}_{k}),{f}^{n-1}\left({x}_{k}),\cdots \hspace{0.33em},f\left({x}_{k}),{x}_{k},\cdots )\in {X}_{f}.p¯k=(pk,pk,pk⋯)∈G¯.{\overline{p}}_{k}=\left({p}_{k},{p}_{k},{p}_{k}\cdots )\in \overline{G}.Then, for any k≥0k\ge 0, we have that d¯(p¯kσ(x¯k),x¯k+1)<∑i=0i=nd(pkfn+1−i(xk),fn−i(xk+1)2i+∑i=n+1∞M2i<∑i=0i=nd(pkfn+1−i(xk),fn−i(xk+1)2i+M2n<∑i=0i=nd(pkfn+1−i(xk),fn−i(xk+1)2i+δ′≤∑i=0i=nd(pkfn+1−i(xk),fn−i(xk+1)2i.\begin{array}{rcl}\overline{d}\left({\overline{p}}_{k}\sigma \left({\overline{x}}_{k}),{\overline{x}}_{k+1})& \lt & \mathop{\displaystyle \sum }\limits_{i=0}^{i=n}\frac{d({p}_{k}{f}^{n+1-i}\left({x}_{k}),{f}^{n-i}\left({x}_{k+1})}{{2}^{i}}+\mathop{\displaystyle \sum }\limits_{i=n+1}^{\infty }\frac{M}{{2}^{i}}\\ & \lt & \mathop{\displaystyle \sum }\limits_{i=0}^{i=n}\frac{d({p}_{k}{f}^{n+1-i}\left({x}_{k}),{f}^{n-i}\left({x}_{k+1})}{{2}^{i}}+\frac{M}{{2}^{n}}\\ & \lt & \mathop{\displaystyle \sum }\limits_{i=0}^{i=n}\frac{d({p}_{k}{f}^{n+1-i}\left({x}_{k}),{f}^{n-i}\left({x}_{k+1})}{{2}^{i}}+{\delta }^{^{\prime} }\\ & \le & \mathop{\displaystyle \sum }\limits_{i=0}^{i=n}\frac{d({p}_{k}{f}^{n+1-i}\left({x}_{k}),{f}^{n-i}\left({x}_{k+1})}{{2}^{i}}.\end{array}According to (9), the definition of Lipschitz map ffand the definition of equivalent map ff, we obtain ∑i=0i=nd(pkfn+1−i(xk),fn−i(xk+1)2i≤∑i=0i=nLn−id(pkf(xk),xk+1)2i≤∑i=0i=nLn−iδ2i=L5δ<ε3.\mathop{\sum }\limits_{i=0}^{i=n}\frac{d({p}_{k}{f}^{n+1-i}\left({x}_{k}),{f}^{n-i}\left({x}_{k+1})}{{2}^{i}}\le \mathop{\sum }\limits_{i=0}^{i=n}\frac{{L}^{n-i}d\left({p}_{k}f\left({x}_{k}),{x}_{k+1})}{{2}^{i}}\le \mathop{\sum }\limits_{i=0}^{i=n}\frac{{L}^{n-i}\delta }{{2}^{i}}={L}^{5}\delta \lt {\varepsilon }_{3}.So for any k≥0k\ge 0, we have d¯(p¯kσ(x¯k),x¯k+1)<L5δ<ε3.\overline{d}\left({\overline{p}}_{k}\sigma \left({\overline{x}}_{k}),{\overline{x}}_{k+1})\lt {L}^{5}\delta \lt {\varepsilon }_{3}.Hence, {xk¯}k=0∞{\left\{\overline{{x}_{k}}\right\}}_{k=0}^{\infty }is (G¯,L5δ)\left(\overline{G},{L}^{5}\delta )-pseudo orbit of the shift map σ\sigma . By the GG-Lipschitz tracking property of the shift map σ\sigma , there exists a point z¯=(z0,z1,z2,⋯zn⋯)∈Xf\overline{z}=\left({z}_{0},{z}_{1},{z}_{2},\cdots {z}_{n}\cdots )\in {X}_{f}such that for every k≥0k\ge 0there exists s¯k=(sk,sk,sk,⋯)∈G¯{\overline{s}}_{k}=\left({s}_{k},{s}_{k},{s}_{k},\cdots )\in \overline{G}with sk∈G{s}_{k}\in Gsuch that d¯(σk(z¯),s¯kx¯k)<L4L5δ.\overline{d}\left({\sigma }^{k}\left(\overline{z}),{\overline{s}}_{k}{\overline{x}}_{k})\lt {L}^{4}{L}^{5}\delta .From the definition of the metric d¯\overline{d}, it follows that d(fk(zn),skxk)<2nL4L5δ.d({f}^{k}\left({z}_{n}),{s}_{k}{x}_{k})\lt {2}^{n}{L}^{4}{L}^{5}\delta .\hspace{0.8em}Thus, d(fk(zn),skxk)<L6δ.d({f}^{k}\left({z}_{n}),{s}_{k}{x}_{k})\lt {L}^{6}\delta .\hspace{0.55em}So the map ffhas the GG-Lipschitz tracking property.□4ConclusionIn this paper, we studied dynamical properties of GG-Lipschitz tracking property and G-asymptotic tracking property. It was obtained that the equivalent conditions of GG-asymptotic tracking property in metric G-space. In addition, it was proved that the self-map ffhas the GG-Lipschitz tracking property if and only if the shift map σ\sigma has the G¯\overline{G}-Lipschitz tracking property in the inverse limit space under topological group action. These results generalize the corresponding results in [Proc. Amer. Math. Soc. 115 (1992), 573–580]. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property

Ji, Zhanjiang
Open Mathematics , Volume 20 (1): 8 – Jan 1, 2022
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de Gruyter
Copyright
© 2022 Zhanjiang Ji, published by De Gruyter
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2391-5455
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2391-5455
DOI
10.1515/math-2022-0026
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Abstract

1IntroductionThe tracking property has an important application in topological dynamical systems. In recent years, more and more scholars pay attention to it, and the relevant research results are shown in [1,2,3, 4,5,6, 7,8,9, 10,11,12, 13,14,15, 16,17]. Liang and Li [1] proved that the self-map ffhas the tracking property if and only if the shift map σ\sigma has the tracking property in the inverse limit space. Ji et al. [2] proved that the shift map has the Lipschitz shadowing property if and only if the self-map has the Lipschitz shadowing property in the inverse limit space. Wang and Zeng [3] gave the relationship between average tracking property and q̲\underline{q}-average tracking property. Wu [4] proved that the self-map ffhas the d¯\overline{d}-tracking property if and only if the shift map σ\sigma has the d¯\overline{d}-tracking property in the inverse limit space.The map ffhas GG-asymptotic tracking property if for each ε>0\varepsilon \gt 0there exists δ>0\delta \gt 0such that for any (G,δ)\left(G,\delta )-pseudo orbit {xi}i≥0{\left\{{x}_{i}\right\}}_{i\ge 0}of ff, and there exists a point y∈Yy\in Yand l≥0l\ge 0such that the sequence {xi}i=l∞{\left\{{x}_{i}\right\}}_{i=l}^{\infty }is (G,ε)\left(G,\varepsilon )shadowed by point yy. We obtained the equivalent condition of the GG-asymptotic tracking property in metric GG-space.The map ffhas GG-Lipschitz tracking property if there exists positive constant LLand δ0{\delta }_{0}such that for any 0<δ<δ00\lt \delta \lt {\delta }_{0}and any (G,δ)\left(G,\delta )-pseudo orbit {xi}i≥0{\left\{{x}_{i}\right\}}_{i\ge 0}of ff, there exists a point x∈Xx\in Xsuch that the sequence {xi}i≥0{\left\{{x}_{i}\right\}}_{i\ge 0}is (G,Lδ)\left(G,L\delta )shadowed by point xx(see [18]). We proved that the map ffhas the GG-Lipschitz tracking property if and only if the shift map σ\sigma has the G¯\overline{G}-Lipschitz tracking property. The main results are as follows in this paper.Theorem 1.1Let (X,d)\left(X,d)be a compact metric G-space, the map f:X→Xf:X\to Xbe an equivalent map and the metric d be invariant to the topological group G, where G is exchangeable. Then, the map f has the G-asymptotic tracking property if and only if for any ε>0\varepsilon \gt 0, there exists 0<δ<ε0\lt \delta \lt \varepsilon and l≥0l\ge 0such that if {xk}k=0∞{\left\{{x}_{k}\right\}}_{k=0}^{\infty }is (G,δ)\left(G,\delta )-pseudo orbit of the map f, then there exists a point y in X such that {fl(xk)}k=0∞{\{{f}^{l}\left({x}_{k})\}}_{k=0}^{\infty }is (G,ε)\left(G,\varepsilon )shadowed by point y.Theorem 1.2Let (X,d)\left(X,d)be a compact metric G-space, (Xf,G¯,d¯,σ)\left({X}_{f},\overline{G},\overline{d},\sigma )be the inverse limit space of (X,G,d,f)\left(X,G,d,f)and the map f:X⟶Xf:X\hspace{0.33em}\longrightarrow \hspace{0.33em}Xbe an equivalent surjection. If the map f is an Lipschitz map with Lipschitz constant L, then we have that the map f has the G-Lipschitz tracking property if and only if the shift map σ\sigma has the G¯\overline{G}-Lipschitz tracking property.2The equivalent condition of GG-asymptotic tracking propertyIn this section, we present some concepts that may be used in the following. The concept of metric GG-space and equivariant map can be found in [17].Definition 2.1[19] Let (X,d)\left(X,d)be a metric space and ffbe a continuous map from XXto XX. The map ffis called to be uniformly continuous if for any ε>0\varepsilon \gt 0there exists 0<δ<ε0\lt \delta \lt \varepsilon such that d(x,y)<δd\left(x,y)\lt \delta implies d(f(x),f(y))<εd(f\left(x),f(y))\lt \varepsilon for all x,y∈Xx,y\in X.Definition 2.2[20] Let (X,d)\left(X,d)be a metric GG-space. The metric ddis said to be invariant to topological group GGprovided that d(gx,gy)=d(x,y)d\left(gx,gy)=d\left(x,y)for all x,y∈Xx,y\in Xand g∈Gg\in G.Definition 2.3Let (X,dX,d) be a metric GG-space and ffbe a continuous map from XXto XX. The map ffhas GG-asymptotic tracking property if for each ε>0\varepsilon \gt 0there exists δ>0\delta \gt 0such that for any (G,δG,\delta )-pseudo orbit {xi}i≥0{\left\{{x}_{i}\right\}}_{i\ge 0}of ff, there exists a point y∈Yy\in Yand l≥0l\ge 0such that the sequence {xi}i=l∞{\left\{{x}_{i}\right\}}_{i=l}^{\infty }is (G,ε)\left(G,\varepsilon )shadowed by point yy.Lemma 2.4Let (X,d)\left(X,d)be a compact metric G-space, the map f:X→Xf:X\to Xbe an equivalent map, the metric d be invariant to the topological group G, where G is exchangeable and m>0m\gt 0. Then, for any ε>0\varepsilon \gt 0, there exists 0<δ<ε0\lt \delta \lt \varepsilon such that if for any k≥0k\ge 0, there exists gk∈G{g}_{k}\in Gsuch that d(gkf(xk),xk+1)<δd\left({g}_{k}f\left({x}_{k}),{x}_{k+1})\lt \delta , then we have d(gk+m−1gk+m−2gk+m−3⋯gk+1gkfm(xk),xm+k)<εd\left({g}_{k+m-1}{g}_{k+m-2}{g}_{k+m-3}\cdots {g}_{k+1}{g}_{k}{f}^{m}\left({x}_{k}),{x}_{m+k})\lt \varepsilon .ProofBy continuity of the map ff, for any ε>0\varepsilon \gt 0and 0≤i<m0\le i\lt m, there exists 0<δ<ε0\lt \delta \lt \varepsilon such that d(x,y)<δd\left(x,y)\lt \delta implies (1)d(fi(x),fi(y))<εm.d({f}^{i}\left(x),{f}^{i}(y))\lt \frac{\varepsilon }{m}.Suppose that for any k>0k\gt 0, there exists gk∈G{g}_{k}\in Gsuch that d(gkf(xk),xk+1)<δ.d\left({g}_{k}f\left({x}_{k}),{x}_{k+1})\lt \delta .\hspace{0.16em}According to the equivalent definition of the map ffand (1), for any k>0k\gt 0and 0≤i<m0\le i\lt m, it follows that d(gkfi+1(xk),fi(xk+1))<εm.d\left({g}_{k}{f}^{i+1}\left({x}_{k}),{f}^{i}\left({x}_{k+1}))\lt \frac{\varepsilon }{m}.\hspace{0.65em}Then, d(gkfm(xk),fm−1(xk+1))<εm.d\left({g}_{k}{f}^{m}\left({x}_{k}),{f}^{m-1}\left({x}_{k+1}))\lt \frac{\varepsilon }{m}.d(gk+1fm−1(xk+1),fm−2(xk+2))<εm.d\left({g}_{k+1}{f}^{m-1}\left({x}_{k+1}),{f}^{m-2}\left({x}_{k+2}))\lt \frac{\varepsilon }{m}.d(gk+2fm−2(xk+2),fm−3(xk+3))<εm.d\left({g}_{k+2}{f}^{m-2}\left({x}_{k+2}),{f}^{m-3}\left({x}_{k+3}))\lt \frac{\varepsilon }{m}.\hspace{0.5em}⋯⋯\cdots \cdots d(gk+m−2f2(xk+m−2),f(xk+m−1))<εm.d\left({g}_{k+m-2}{f}^{2}\left({x}_{k+m-2}),f\left({x}_{k+m-1}))\lt \frac{\varepsilon }{m}.d(gk+m−1f(xk+m−1),xk+m)<εm.d\left({g}_{k+m-1}f\left({x}_{k+m-1}),{x}_{k+m})\lt \frac{\varepsilon }{m}.Since the metric ddis invariant to the topological group GGand GGis exchangeable, we have d(gkgk+1gk+2⋯gk+m−1fm(xk),gk+1gk+2⋯gk+m−1fm−1(xk+1))<εm.d\left({g}_{k}{g}_{k+1}{g}_{k+2}\cdots {g}_{k+m-1}{f}^{m}\left({x}_{k}),{g}_{k+1}{g}_{k+2}\cdots {g}_{k+m-1}{f}^{m-1}\left({x}_{k+1}))\lt \frac{\varepsilon }{m}.d(gk+1gk+2gk+3⋯gk+m−1fm−1(xk+1),gk+2gk+3⋯gk+m−1fm−2(xk+2))<εm.d\left({g}_{k+1}{g}_{k+2}{g}_{k+3}\cdots {g}_{k+m-1}{f}^{m-1}\left({x}_{k+1}),{g}_{k+2}{g}_{k+3}\cdots {g}_{k+m-1}{f}^{m-2}\left({x}_{k+2}))\lt \frac{\varepsilon }{m}.d(gk+2gk+3⋯gk+m−1fm−2(xk+2),gk+3⋯gk+m−1fm−3(xk+3))<εm.d\left({g}_{k+2}{g}_{k+3}\cdots {g}_{k+m-1}{f}^{m-2}\left({x}_{k+2}),{g}_{k+3}\cdots {g}_{k+m-1}{f}^{m-3}\left({x}_{k+3}))\lt \frac{\varepsilon }{m}.⋯⋯\cdots \cdots d(gk+m−2gk+m−1f2(xk+m−2),gk+m−1f(xk+m−1))<εm.d\left({g}_{k+m-2}{g}_{k+m-1}{f}^{2}\left({x}_{k+m-2}),{g}_{k+m-1}f\left({x}_{k+m-1}))\lt \frac{\varepsilon }{m}.d(gk+m−1f(xk+m−1),xk+m)<εm.d\left({g}_{k+m-1}f\left({x}_{k+m-1}),{x}_{k+m})\lt \frac{\varepsilon }{m}.Therefore, d(gkgk+1gk+2⋯gk+m−1fm(xk),xm+k)<d(gkgk+1gk+2⋯gk+m−1fm(xk),gk+1gk+2⋯gk+m−1fm−1(xk+1))+d(gk+1gk+2gk+3⋯gk+m−1fm−1(xk+1),gk+2gk+3⋯gk+m−1fm−2(xk+2))+d(gk+2gk+3⋯gk+m−1fm−2(xk+2),gk+3⋯gk+m−1fm−3(xk+3))+⋯⋯+d(gk+m−2gk+m−1f2(xk+m−2),gk+m−1f(xk+m−1))+d(gk+m−1f(xk+m−1),xk+m)<εm+εm+εm+⋯⋯+εm=ε.□\hspace{5.9em}\begin{array}{l}d\left({g}_{k}{g}_{k+1}{g}_{k+2}\cdots {g}_{k+m-1}{f}^{m}\left({x}_{k}),{x}_{m+k})\\ \hspace{1.0em}\lt d\left({g}_{k}{g}_{k+1}{g}_{k+2}\cdots {g}_{k+m-1}{f}^{m}\left({x}_{k}),{g}_{k+1}{g}_{k+2}\cdots {g}_{k+m-1}{f}^{m-1}\left({x}_{k+1}))\\ \hspace{2.0em}+d\left({g}_{k+1}{g}_{k+2}{g}_{k+3}\cdots {g}_{k+m-1}{f}^{m-1}\left({x}_{k+1}),{g}_{k+2}{g}_{k+3}\cdots {g}_{k+m-1}{f}^{m-2}\left({x}_{k+2}))\hspace{2.0em}+d\left({g}_{k+2}{g}_{k+3}\cdots {g}_{k+m-1}{f}^{m-2}\left({x}_{k+2}),{g}_{k+3}\cdots {g}_{k+m-1}{f}^{m-3}\left({x}_{k+3}))\\ \hspace{1.0em}\hspace{1.0em}+\cdots \cdots +d\left({g}_{k+m-2}{g}_{k+m-1}{f}^{2}\left({x}_{k+m-2}),{g}_{k+m-1}f\left({x}_{k+m-1}))+d\left({g}_{k+m-1}f\left({x}_{k+m-1}),{x}_{k+m})\\ \hspace{1.0em}\lt \frac{\varepsilon }{m}+\frac{\varepsilon }{m}+\frac{\varepsilon }{m}+\cdots \cdots +\frac{\varepsilon }{m}=\varepsilon .\end{array}\hspace{3.75em}<mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/"xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">\square </mml:mpadded>Theorem 2.5Let (X,d)\left(X,d)be a compact metric G-space, the map f:X→Xf:X\to Xbe an equivalent map and the metric d be invariant to the topological group G where G is exchangeable. Then, the map f has the G-asymptotic tracking property if and only if for any ε>0\varepsilon \gt 0there exists 0<δ<ε0\lt \delta \lt \varepsilon and l≥0l\ge 0such that if {xk}k=0∞{\left\{{x}_{k}\right\}}_{k=0}^{\infty }is (G,δ)\left(G,\delta )-pseudo orbit of the map f, then there exists a point y in X such that {fl(xk)}k=0∞{\{{f}^{l}\left({x}_{k})\}}_{k=0}^{\infty }is (G,ε)\left(G,\varepsilon )shadowed by point y.Proof(Necessity) Suppose that the map ffhas the GG-asymptotic tracking property. Then, for each ε>0\varepsilon \gt 0, there exists 0<τ<ε20\lt \tau \lt \frac{\varepsilon }{2}such that for any (G,τ)\left(G,\tau )-pseudo orbit {xk}k≥0{\left\{{x}_{k}\right\}}_{k\ge 0}of ff, there exists a point y∈Xy\in Xand l≥0l\ge 0such that the sequence {xk}k=l∞{\left\{{x}_{k}\right\}}_{k=l}^{\infty }is (G,ε2)\left(G,\frac{\varepsilon }{2})shadowed by point yy. If l=0l=0, the results are obvious. Now we assume l>0l\gt 0. Since the map ffis uniformly continuous, for given ε2l>0\frac{\varepsilon }{2l}\gt 0and any 0≤i<l0\le i\lt l, there exists 0<δ<minε2l,τ0\lt \delta \lt \min \left\{\frac{\varepsilon }{2l},\tau \right\}such that d(x,y)<δd\left(x,y)\lt \delta implies (2)d(fi(x),fi(y))<ε2l.d({f}^{i}\left(x),{f}^{i}(y))\lt \frac{\varepsilon }{2l}.Let {xk}k=0∞{\left\{{x}_{k}\right\}}_{k=0}^{\infty }be (G,δ)\left(G,\delta )-pseudo orbit of the map ff. Then, for any k≥0k\ge 0, there exists tk∈G{t}_{k}\in Gsuch that d(tkf(xk),xk+1)<δ.d\left({t}_{k}f\left({x}_{k}),{x}_{k+1})\lt \delta .\hspace{0.16em}By (2) and the equivalent definition of the map ff, for any k≥0k\ge 0and 0≤i<l0\le i\lt l, we have that (3)d(tkfi+1(xk),fi(xk+1))<ε2l.d\left({t}_{k}{f}^{i+1}\left({x}_{k}),{f}^{i}\left({x}_{k+1}))\lt \frac{\varepsilon }{2l}.Noting that the metric ddis invariant to the topological group GG, where GGis exchangeable and (3), we have d(tktk+1tk+2⋯tk+l−2tk+l−1fl(xk),tk+1tk+2⋯tk+l−2tk+l−1fl−1(xk+1))<ε2l,d\left({t}_{k}{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l}\left({x}_{k}),{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-1}\left({x}_{k+1}))\lt \frac{\varepsilon }{2l},d(tk+1tk+2tk+3⋯tk+l−2tk+l−1fl−1(xk+1),tk+2tk+3⋯tk+l−2tk+l−1fl−2(xk+2))<ε2l,d\left({t}_{k+1}{t}_{k+2}{t}_{k+3}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-1}\left({x}_{k+1}),{t}_{k+2}{t}_{k+3}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-2}\left({x}_{k+2}))\lt \frac{\varepsilon }{2l},d(tk+2tk+3⋯tk+l−2tk+l−1fl−2(xk+2),tk+3⋯tk+l−2tk+l−1fl−3(xk+3))<ε2l,d\left({t}_{k+2}{t}_{k+3}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-2}\left({x}_{k+2}),{t}_{k+3}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-3}\left({x}_{k+3}))\lt \frac{\varepsilon }{2l},⋯⋯\cdots \cdots d(tk+l−2tk+l−1f2(xk+l−2),tk+l−1f(xk+l−1))<ε2l,d\left({t}_{k+l-2}{t}_{k+l-1}{f}^{2}\left({x}_{k+l-2}),{t}_{k+l-1}f\left({x}_{k+l-1}))\lt \frac{\varepsilon }{2l},d(tk+l−1f(xk+l−1),xk+l)<ε2l,d\left({t}_{k+l-1}f\left({x}_{k+l-1}),{x}_{k+l})\lt \frac{\varepsilon }{2l},and thus, d(tktk+1tk+2⋯tk+l−2tk+l−1fl(xk),xk+l)<d(tktk+1tk+2⋯tk+l−2tk+l−1fl(xk),tk+1tk+2⋯tk+l−2tk+l−1fl−1(xk+1))+d(tk+1tk+2tk+3⋯tk+l−2tk+l−1fl−1(xk+1),tk+2tk+3⋯tk+l−2tk+l−1fl−2(xk+2))+d(tk+2tk+3⋯tk+l−2tk+l−1fl−2(xk+2),tk+3⋯tk+l−2tk+l−1fl−3(xk+3))+⋯⋯+d(tk+l−2tk+l−1f2(xk+l−2),tk+l−1f(xk+l−1))+d(tk+l−1f(xk+l−1),xk+l)<ε2l+ε2l+ε2l+⋯⋯+ε2l=ε2.\begin{array}{l}d\left({t}_{k}{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l}\left({x}_{k}),{x}_{k+l})\\ \hspace{1.0em}\lt d\left({t}_{k}{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l}\left({x}_{k}),{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-1}\left({x}_{k+1}))\\ \hspace{2.0em}+d\left({t}_{k+1}{t}_{k+2}{t}_{k+3}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-1}\left({x}_{k+1}),{t}_{k+2}{t}_{k+3}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-2}\left({x}_{k+2}))\hspace{2.0em}+d\left({t}_{k+2}{t}_{k+3}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-2}\left({x}_{k+2}),{t}_{k+3}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l-3}\left({x}_{k+3}))\\ \hspace{2.0em}+\cdots \cdots +d\left({t}_{k+l-2}{t}_{k+l-1}{f}^{2}\left({x}_{k+l-2}),{t}_{k+l-1}f\left({x}_{k+l-1}))+d\left({t}_{k+l-1}f\left({x}_{k+l-1}),{x}_{k+l})\\ \hspace{1.0em}\lt \frac{\varepsilon }{2l}+\frac{\varepsilon }{2l}+\frac{\varepsilon }{2l}+\cdots \cdots +\frac{\varepsilon }{2l}=\frac{\varepsilon }{2}.\end{array}So for any k≥0k\ge 0, we have (4)d(tktk+1tk+2⋯tk+l−2tk+l−1fl(xk),xk+l)<ε2.d\left({t}_{k}{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l}\left({x}_{k}),{x}_{k+l})\lt \frac{\varepsilon }{2}.\hspace{1.1em}Since the map ffhas the GG-asymptotic tracking property, for any k≥0k\ge 0, there exists gk∈G{g}_{k}\in Gand y∈Yy\in Ysuch that d(fk(y),gkxk+l)<ε2.d({f}^{k}(y),{g}_{k}{x}_{k+l})\lt \frac{\varepsilon }{2}.\hspace{2.7em}Since the metric ddis invariant to the topological group GG, then (5)d(gk−1fk(y),xk+l)<ε2.d\left({g}_{k}^{-1}{f}^{k}(y),{x}_{k+l})\lt \frac{\varepsilon }{2}.\hspace{2.5em}By (4) and (5), for any k≥0k\ge 0, we obtain d(tktk+1tk+2⋯tk+l−2tk+l−1fl(xk),gk−1fk(y))<d(tktk+1tk+2⋯tk+l−2tk+l−1fl(xk),xk+l)+d(xk+l,gk−1fk(y))<ε.d\left({t}_{k}{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l}\left({x}_{k}),{g}_{k}^{-1}{f}^{k}(y))\lt d\left({t}_{k}{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l}\left({x}_{k}),{x}_{k+l})+d\left({x}_{k+l},{g}_{k}^{-1}{f}^{k}(y))\lt \varepsilon .Together with the fact that the metric ddis invariant to the topological group GGagain, it follows that d(fk(y),gktktk+1tk+2⋯tk+l−2tk+l−1fl(xk))<ε.d({f}^{k}(y),{g}_{k}{t}_{k}{t}_{k+1}{t}_{k+2}\cdots {t}_{k+l-2}{t}_{k+l-1}{f}^{l}\left({x}_{k}))\lt \varepsilon .Hence, the sequence {fl(xk)}k=0∞{\{{f}^{l}\left({x}_{k})\}}_{k=0}^{\infty }is (G,ε)\left(G,\varepsilon )shadowed by point yy.(Sufficiency) Suppose that for any ε>0\varepsilon \gt 0there exists 0<τ<ε0\lt \tau \lt \varepsilon and l≥0l\ge 0such that if {xk}k=0∞{\left\{{x}_{k}\right\}}_{k=0}^{\infty }is (G,τ)\left(G,\tau )-pseudo orbit of the map ff, then there exists a point zzin XXsuch that {fl(xk)}k=0∞{\{{f}^{l}\left({x}_{k})\}}_{k=0}^{\infty }is (G,ε2)\left(G,\frac{\varepsilon }{2}\right)shadowed by point zz. If l=0l=0, the results are obvious. Now we assume l>0l\gt 0. Since the map ffis uniformly continuous, for given ε>0\varepsilon \gt 0and any 0≤i<l0\le i\lt l, there exists 0<δ0<τ0\lt {\delta }_{0}\lt \tau such that d(x,y)<δd\left(x,y)\lt \delta implies d(fi(x),fi(y))<ε2.d({f}^{i}\left(x),{f}^{i}(y))\lt \frac{\varepsilon }{2}.Let {xk}k=0∞{\{{x}_{k}\}}_{k=0}^{\infty }be (G,δ0)\left(G,{\delta }_{0})-pseudo orbit of the map ff. Then, there exists z∈Xz\in Xsuch that for any k≥0k\ge 0there exists pk∈G{p}_{k}\in Gsuch that (6)dfk(z),pkfl(xk))<ε2.d\left({f}^{k}\left(z)\left,{p}_{k}{f}^{l}\left({x}_{k})\right)\lt \frac{\varepsilon }{2}.\right.In addition, for any k≥0k\ge 0, there exists sk∈G{s}_{k}\in Gsuch that d(skf(xk),xk+1)<δ0.d\left({s}_{k}f\left({x}_{k}),{x}_{k+1})\lt {\delta }_{0}.\hspace{1em}By Lemma 2.9, for any k≥0k\ge 0, we have that (7)d(sksk+1sk+2⋯sk+l−2sk+l−1fl(xk),xk+l)<ε2.d\left({s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{f}^{l}\left({x}_{k}),{x}_{k+l})\lt \frac{\varepsilon }{2}.Since the metric ddis invariant to the topological group GGand equations (6) and (7), we have dsksk+1sk+2⋯sk+l−2sk+l−1fk(z),sksk+1sk+2⋯sk+l−2sk+l−1pkfl(xk))<ε2d\left({s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{f}^{k}\left(z),{s}_{k}{s}_{k+1}{s}_{k+2}\left\cdots {s}_{k+l-2}{s}_{k+l-1}{p}_{k}{f}^{l}\left({x}_{k})\right)\lt \frac{\varepsilon }{2}\right.and d(pksksk+1sk+2⋯sk+l−2sk+l−1f(xk),pkxk+l)<ε2.d\left({p}_{k}{s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}f\left({x}_{k}),{p}_{k}{x}_{k+l})\lt \frac{\varepsilon }{2}.Since GGis exchangeable, we have that d(sksk+1sk+2⋯sk+l−2sk+l−1fk(z),pkxk+l)<d(sksk+1sk+2⋯sk+l−2sk+l−1fk(z),sksk+1sk+2⋯sk+l−2sk+l−1pkfl(xk)+d(sksk+1sk+2⋯sk+l−2sk+l−1pkfl(xk),pkxk+l)=d(sksk+1sk+2⋯sk+l−2sk+l−1fk(z),sksk+1sk+2⋯sk+l−2sk+l−1pkfl(xk)+d(pksksk+1sk+2⋯sk+l−2sk+l−1fl(xk),pkxk+1)<ε2+ε2<ε.\begin{array}{l}d\left({s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{f}^{k}\left(z),{p}_{k}{x}_{k+l})\\ \hspace{1.0em}\lt d({s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{f}^{k}\left(z),{s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{p}_{k}{f}^{l}\left({x}_{k})+d\left({s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{p}_{k}{f}^{l}\left({x}_{k}),{p}_{k}{x}_{k+l})\\ \hspace{1.0em}=d({s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{f}^{k}\left(z),{s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{p}_{k}{f}^{l}\left({x}_{k})+d\left({p}_{k}{s}_{k}{s}_{k+1}{s}_{k+2}\cdots {s}_{k+l-2}{s}_{k+l-1}{f}^{l}\left({x}_{k}),{p}_{k}{x}_{k+1})\\ \hspace{1.0em}\lt \frac{\varepsilon }{2}+\frac{\varepsilon }{2}\lt \varepsilon .\end{array}\hspace{0.25em}Hence, the map ffhas the GG-asymptotic tracking property. Thus, we complete the proof.□3GG-Lipschitz tracking propertyThe concept of the inverse limit spaces in this section under group action can be found in [21].Definition 3.1[5] Let(X,d)\left(X,d)be a metric space and f:X→Xf:X\to Xbe a continuous map.The map ffis said to be an Lipschitz map if there exists a positive constant L>0L\gt 0such that d(f(x),f(y))≤Ld(x,y)d(f\left(x),f(y))\le Ld\left(x,y)for all x,y∈Xx,y\in X.Definition 3.2[18] Let(X,d)\left(X,d)be a metric GG-space and f:X→Xf:X\to Xbe a continuous map. The map ffhas GG-Lipschitz tracking property if there exists positive constant LLand δ0{\delta }_{0}such that for any 0<δ<δ00\lt \delta \lt {\delta }_{0}and (G,δ)\left(G,\delta )-pseudo orbit {xi}i≥0{\left\{{x}_{i}\right\}}_{i\ge 0}of ffthere exists a point zzin XXsuch that the sequence {xi}i≥0{\left\{{x}_{i}\right\}}_{i\ge 0}is (G,Lδ)\left(G,L\delta )shadowed by point zz.Now, we give the proof of Theorem 3.3.Theorem 3.3Let (X,d)\left(X,d)be a compact metric G-space, (Xf,G¯,d¯,σ)\left({X}_{f},\overline{G},\overline{d},\sigma )be the inverse limit space of (X,G,d,f)\left(X,G,d,f)and the map f:X⟶Xf:X\hspace{0.33em}\longrightarrow \hspace{0.33em}Xbe an equivalent surjection. If the map f is an Lipschitz map with Lipschitz constant L, then we have that the map f has the G-Lipschitz tracking property if and only if the shift map σ\sigma has the G¯\overline{G}-Lipschitz tracking property.Proof(Necessity) Suppose that the map ffhas the GG-Lipschitz tracking property. Then, there exists positive constant L1{L}_{1}and ε1{\varepsilon }_{1}such that for any 0<ε<ε10\lt \varepsilon \lt {\varepsilon }_{1}and (G,ε)\left(G,\varepsilon )-pseudo orbit {xi}i=0∞{\left\{{x}_{i}\right\}}_{i=0}^{\infty }of the map ff, there exists a point x∈Xx\in Xsuch that {xi}i=0∞{\left\{{x}_{i}\right\}}_{i=0}^{\infty }is (G,L1ε)\left(G,{L}_{1}\varepsilon )shadowed by xx. By the compactness of XX, let M=diam(X)M={\rm{diam}}\left(X). For given ε>0\varepsilon \gt 0, choose a positive constant m>0m\gt 0such that M2m<ε\frac{M}{{2}^{m}}\lt \varepsilon . Let L2=Lm+Lm−12+Lm−222+⋯+L2m−1+12m,{L}_{2}={L}_{m}+\frac{{L}_{m-1}}{2}+\frac{{L}_{m-2}}{{2}^{2}}+\cdots +\frac{L}{{2}^{m-1}}+\frac{1}{{2}^{m}},L3=L1L22m,{L}_{3}={L}_{1}{L}_{2}{2}^{m},ε2=ε12m.{\varepsilon }_{2}=\frac{{\varepsilon }_{1}}{{2}^{m}}.For any 0<η<ε20\lt \eta \lt {\varepsilon }_{2}, let {y¯k}k=0∞=(yk0,yk1,yk2⋯){\{{\overline{y}}_{k}\}}_{k=0}^{\infty }=({y}_{k}^{0},{y}_{k}^{1},{y}_{k}^{2}\cdots )be (G,η)\left(G,\eta )-pseudo orbit of the shift map σ\sigma in Xf{X}_{f}. It is obvious that for each k≥0k\ge 0, there exists g¯k=(gk,gk,gk⋯)∈G¯{\overline{g}}_{k}=\left({g}_{k},{g}_{k},{g}_{k}\cdots )\in \overline{G}such that d¯(g¯kσ(y¯k),y¯k+1)<η.\overline{d}\left({\overline{g}}_{k}\sigma ({\overline{y}}_{k}),{\overline{y}}_{k+1})\lt \eta .From the definition of the metric d¯\overline{d}, for every k≥0k\ge 0, it follows that d(gkf(ykm),yk+1m)<2mη<ε1.\hspace{5.75em}d\left({g}_{k}f({y}_{k}^{m}),{y}_{k+1}^{m})\lt {2}^{m}\eta \lt {\varepsilon }_{1}.\hspace{6.1em}Thus, {ykm}k=0∞{\{{y}_{k}^{m}\}}_{k=0}^{\infty }is (G,2mη)\left(G,{2}^{m}\eta )-pseudo orbit of the map ff. By the GG-Lipschitz tracking property of the map ffin XX, there exists a point yyin XXsuch that for every k≥0k\ge 0there exists tk∈G{t}_{k}\in Gsuch that (8)d(fk(y),tkykm)<L12mη.d({f}^{k}(y),{t}_{k}{y}_{k}^{m})\lt {L}_{1}{2}^{m}\eta .Since the map ffis onto, we write y¯=(fm(y),fm−1(y),⋯,f(y),y,⋯)∈Xf.\hspace{5.85em}\overline{y}=({f}^{m}(y),{f}^{m-1}(y),\cdots \hspace{0.33em},f(y),y,\cdots )\in {X}_{f}.\hspace{5.85em}t¯k=(tk,tk,tk⋯)∈G¯.{\overline{t}}_{k}=\left({t}_{k},{t}_{k},{t}_{k}\cdots )\in \overline{G}.\hspace{5.85em}By the definition of the equivalent map ff, for any k≥0k\ge 0, we have d¯(σk(y¯),t¯ky¯k)<∑i=0i=md(fk+m−i(y),tkyki)2i+∑i=m+1∞M2i<∑i=0i=md(fm−i(fk(y)),fm−i(tkykm))2i+M2m<∑i=0i=md(fm−i(fk(y)),fm−i(tkykm))2i+ε≤∑i=0i=md(fm−i(fk(y)),fm−i(tkykm))2i.\begin{array}{rcl}\overline{d}\left({\sigma }^{k}(\overline{y}),{\overline{t}}_{k}{\overline{y}}_{k})& \lt & \mathop{\displaystyle \sum }\limits_{i=0}^{i=m}\frac{d({f}^{k+m-i}(y),{t}_{k}{y}_{k}^{i})}{{2}^{i}}+\mathop{\displaystyle \sum }\limits_{i=m+1}^{\infty }\frac{M}{{2}^{i}}\\ & \lt & \mathop{\displaystyle \sum }\limits_{i=0}^{i=m}\frac{d({f}^{m-i}({f}^{k}(y)),{f}^{m-i}\left({t}_{k}{y}_{k}^{m}))}{{2}^{i}}+\frac{M}{{2}^{m}}\\ & \lt & \mathop{\displaystyle \sum }\limits_{i=0}^{i=m}\frac{d({f}^{m-i}({f}^{k}(y)),{f}^{m-i}\left({t}_{k}{y}_{k}^{m}))}{{2}^{i}}+\varepsilon \\ & \le & \mathop{\displaystyle \sum }\limits_{i=0}^{i=m}\frac{d({f}^{m-i}({f}^{k}(y)),{f}^{m-i}\left({t}_{k}{y}_{k}^{m}))}{{2}^{i}}.\end{array}According to (8) and the definition of Lipschitz map ff, we obtain ∑i=0i=md(fm−i(fk(y)),fm−i(tkykm))2i<∑i=0i=mLm−id(fk(y),tkykm)2i<∑i=0i=mLm−iL12mη2i=L1L22mη.\mathop{\sum }\limits_{i=0}^{i=m}\frac{d({f}^{m-i}({f}^{k}(y)),{f}^{m-i}\left({t}_{k}{y}_{k}^{m}))}{{2}^{i}}\lt \mathop{\sum }\limits_{i=0}^{i=m}\frac{{L}^{m-i}d({f}^{k}(y),{t}_{k}{y}_{k}^{m})}{{2}^{i}}\lt \mathop{\sum }\limits_{i=0}^{i=m}\frac{{L}^{m-i}{L}_{1}{2}^{m}\eta }{{2}^{i}}={L}_{1}{L}_{2}{2}^{m}\eta .Then, for any k≥0k\ge 0, it follows that d¯(σk(y¯),t¯ky¯k)<L3η.\overline{d}\left({\sigma }^{k}(\overline{y}),{\overline{t}}_{k}{\overline{y}}_{k})\lt {L}_{3}\eta .Therefore, the shift map σ\sigma has the GG-Lipschitz tracking property.(Sufficiency) Next, we suppose that the shift map σ\sigma has the G¯\overline{G}-Lipschitz tracking property. Then, there exists positive constant L4{L}_{4}and ε3{\varepsilon }_{3}such that for any 0<δ′<ε30\lt {\delta }^{^{\prime} }\lt {\varepsilon }_{3}and (G¯,δ′)\left(\overline{G},{\delta }^{^{\prime} })-pseudo orbit {z¯k}k=0∞{\left\{{\overline{z}}_{k}\right\}}_{k=0}^{\infty }of the shift map σ\sigma there exists a point z¯∈Xf\overline{z}\in {X}_{f}such that {z¯k}k=0∞{\left\{{\overline{z}}_{k}\right\}}_{k=0}^{\infty }is (G¯,L4δ′)\left(\overline{G},{L}_{4}{\delta }^{^{\prime} })shadowed by point z¯.\overline{z}.For given δ′>0{\delta }^{^{\prime} }\gt 0, choose n>0n\gt 0such that M2n<δ′\frac{M}{{2}^{n}}\lt {\delta }^{^{\prime} }. Then, we write L5=Ln+Ln−12Ln−222+⋯+L2n−1+12n,{L}_{5}={L}_{n}+\frac{{L}_{n-1}}{2}\frac{{L}_{n-2}}{{2}^{2}}+\cdots +\frac{L}{{2}^{n-1}}+\frac{1}{{2}^{n}},L6=2nL4L5,{L}_{6}={2}^{n}{L}_{4}{L}_{5},ε4=ε3L5.{\varepsilon }_{4}=\frac{{\varepsilon }_{3}}{{L}_{5}}.For any 0<δ<ε40\lt \delta \lt {\varepsilon }_{4}, suppose that {xk}k=0∞{\left\{{x}_{k}\right\}}_{k=0}^{\infty }is (G,δ)\left(G,\delta )-pseudo orbit of the map ff. It is obvious that for each k≥0k\ge 0, there exists a point pk∈G{p}_{k}\in Gsuch that (9)d(pkf(xk),xk+1)<δ.d\left({p}_{k}f\left({x}_{k}),{x}_{k+1})\lt \delta .\hspace{1.7em}Since the map ffis onto, we write x¯k=(fn(xk),fn−1(xk),⋯,f(xk),xk,⋯)∈Xf.{\overline{x}}_{k}=({f}^{n}\left({x}_{k}),{f}^{n-1}\left({x}_{k}),\cdots \hspace{0.33em},f\left({x}_{k}),{x}_{k},\cdots )\in {X}_{f}.p¯k=(pk,pk,pk⋯)∈G¯.{\overline{p}}_{k}=\left({p}_{k},{p}_{k},{p}_{k}\cdots )\in \overline{G}.Then, for any k≥0k\ge 0, we have that d¯(p¯kσ(x¯k),x¯k+1)<∑i=0i=nd(pkfn+1−i(xk),fn−i(xk+1)2i+∑i=n+1∞M2i<∑i=0i=nd(pkfn+1−i(xk),fn−i(xk+1)2i+M2n<∑i=0i=nd(pkfn+1−i(xk),fn−i(xk+1)2i+δ′≤∑i=0i=nd(pkfn+1−i(xk),fn−i(xk+1)2i.\begin{array}{rcl}\overline{d}\left({\overline{p}}_{k}\sigma \left({\overline{x}}_{k}),{\overline{x}}_{k+1})& \lt & \mathop{\displaystyle \sum }\limits_{i=0}^{i=n}\frac{d({p}_{k}{f}^{n+1-i}\left({x}_{k}),{f}^{n-i}\left({x}_{k+1})}{{2}^{i}}+\mathop{\displaystyle \sum }\limits_{i=n+1}^{\infty }\frac{M}{{2}^{i}}\\ & \lt & \mathop{\displaystyle \sum }\limits_{i=0}^{i=n}\frac{d({p}_{k}{f}^{n+1-i}\left({x}_{k}),{f}^{n-i}\left({x}_{k+1})}{{2}^{i}}+\frac{M}{{2}^{n}}\\ & \lt & \mathop{\displaystyle \sum }\limits_{i=0}^{i=n}\frac{d({p}_{k}{f}^{n+1-i}\left({x}_{k}),{f}^{n-i}\left({x}_{k+1})}{{2}^{i}}+{\delta }^{^{\prime} }\\ & \le & \mathop{\displaystyle \sum }\limits_{i=0}^{i=n}\frac{d({p}_{k}{f}^{n+1-i}\left({x}_{k}),{f}^{n-i}\left({x}_{k+1})}{{2}^{i}}.\end{array}According to (9), the definition of Lipschitz map ffand the definition of equivalent map ff, we obtain ∑i=0i=nd(pkfn+1−i(xk),fn−i(xk+1)2i≤∑i=0i=nLn−id(pkf(xk),xk+1)2i≤∑i=0i=nLn−iδ2i=L5δ<ε3.\mathop{\sum }\limits_{i=0}^{i=n}\frac{d({p}_{k}{f}^{n+1-i}\left({x}_{k}),{f}^{n-i}\left({x}_{k+1})}{{2}^{i}}\le \mathop{\sum }\limits_{i=0}^{i=n}\frac{{L}^{n-i}d\left({p}_{k}f\left({x}_{k}),{x}_{k+1})}{{2}^{i}}\le \mathop{\sum }\limits_{i=0}^{i=n}\frac{{L}^{n-i}\delta }{{2}^{i}}={L}^{5}\delta \lt {\varepsilon }_{3}.So for any k≥0k\ge 0, we have d¯(p¯kσ(x¯k),x¯k+1)<L5δ<ε3.\overline{d}\left({\overline{p}}_{k}\sigma \left({\overline{x}}_{k}),{\overline{x}}_{k+1})\lt {L}^{5}\delta \lt {\varepsilon }_{3}.Hence, {xk¯}k=0∞{\left\{\overline{{x}_{k}}\right\}}_{k=0}^{\infty }is (G¯,L5δ)\left(\overline{G},{L}^{5}\delta )-pseudo orbit of the shift map σ\sigma . By the GG-Lipschitz tracking property of the shift map σ\sigma , there exists a point z¯=(z0,z1,z2,⋯zn⋯)∈Xf\overline{z}=\left({z}_{0},{z}_{1},{z}_{2},\cdots {z}_{n}\cdots )\in {X}_{f}such that for every k≥0k\ge 0there exists s¯k=(sk,sk,sk,⋯)∈G¯{\overline{s}}_{k}=\left({s}_{k},{s}_{k},{s}_{k},\cdots )\in \overline{G}with sk∈G{s}_{k}\in Gsuch that d¯(σk(z¯),s¯kx¯k)<L4L5δ.\overline{d}\left({\sigma }^{k}\left(\overline{z}),{\overline{s}}_{k}{\overline{x}}_{k})\lt {L}^{4}{L}^{5}\delta .From the definition of the metric d¯\overline{d}, it follows that d(fk(zn),skxk)<2nL4L5δ.d({f}^{k}\left({z}_{n}),{s}_{k}{x}_{k})\lt {2}^{n}{L}^{4}{L}^{5}\delta .\hspace{0.8em}Thus, d(fk(zn),skxk)<L6δ.d({f}^{k}\left({z}_{n}),{s}_{k}{x}_{k})\lt {L}^{6}\delta .\hspace{0.55em}So the map ffhas the GG-Lipschitz tracking property.□4ConclusionIn this paper, we studied dynamical properties of GG-Lipschitz tracking property and G-asymptotic tracking property. It was obtained that the equivalent conditions of GG-asymptotic tracking property in metric G-space. In addition, it was proved that the self-map ffhas the GG-Lipschitz tracking property if and only if the shift map σ\sigma has the G¯\overline{G}-Lipschitz tracking property in the inverse limit space under topological group action. These results generalize the corresponding results in [Proc. Amer. Math. Soc. 115 (1992), 573–580].

Journal

Open Mathematics – de Gruyter

Published: Jan 1, 2022

Keywords: metric G -space; topological group; inverse limit space; G -Lipschitz tracking property; G -asymptotic tracking property; 37B99

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