On Non-visibility of Kobayashi Geodesics and the Geometry of the Boundary

We show that on convex domains with C1,alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}<^>{1,\alpha }$$\end{document}-smooth boundary the limit set of non-visible Kobayashi geodesics are contained in a complex face. In C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}<^>2$$\end{document} this implies the existence of a complex tangential line segment of non-Goldilocks point in the boundary. Conversely, we construct sequences of non-visible almost-geodesics on (C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}$$\end{document}-)convex domains whose boundary contains a complex tangential line segment of non-Goldilocks points in a specific direction.