Quasi-Shadowing for Partially Hyperbolic Flows with a Local Product Structure

Let a flow ϕt be partially hyperbolic on Λ. If Λ has a local product structure, then ϕt has the quasi-shadowing property on Λ in the following sense: for any 𝜖 > 0, there exists constant δ > 0 such that for any (δ,1)-pseudo orbit {xk,tk}k∈ℤ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{x_{k}, t_{k}\}_{k\in \mathbb {Z}}$\end{document} of ϕt with 1 ≤ tk ≤ 2 for all k∈ℤ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k\in \mathbb {Z}$\end{document}, there exist a sequence of points {yk}k∈ℤ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{y_{k}\}_{k\in \mathbb {Z}}$\end{document} and a reparametrization α∈Rep(ℝ,𝜖)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha \in Rep(\mathbb {R},\epsilon )$\end{document} such that ϕα(t)−α(Σk)(yk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\phi _{\alpha (t)-\alpha ({\Sigma }_{k})}(y_{k})$\end{document} trace ϕt−Σk(xk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\phi _{t-{\Sigma }_{k}}(x_{k})$\end{document} in which yk+ 1 lies in the local center leaf of ϕα(Σk+1)−α(Σk)(yk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\phi _{\alpha ({\Sigma }_{k+1})-\alpha ({\Sigma }_{k})}(y_{k})$\end{document} for k ≥ 0 t ≥ 0 and ϕα(t)−α(−Σk)(yk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\phi _{\alpha (t)-\alpha (-{\Sigma }_{k})}(y_{k})$\end{document} trace ϕt−(−Σk)(xk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\phi _{t-(-{\Sigma }_{k})}(x_{k})$\end{document} in which yk+ 1 lies in the local center leaf of ϕα(−Σk+1)−α(−Σk)(yk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\phi _{\alpha (-{\Sigma }_{k+1})-\alpha (-{\Sigma }_{k})}(y_{k})$\end{document} for k < 0, t < 0.