Lyapunov Exponents Everywhere and Rigidity

In the present work, we obtain rigidity results analyzing the set of regular points, in the sense of Oseledec’s Theorem. It is presented a study on the possibility of Anosov diffeomorphisms having all Lyapunov exponents defined everywhere. We prove that this condition implies local rigidity of an Anosov automorphism of the torus Td,d≥3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {T}^{d}, d \geq 3,$\end{document}C1 −close to a linear automorphism diagonalizable over ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}$\end{document} and such that its characteristic polynomial is irreducible over ℚ.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {Q}.$\end{document}