\(\sigma\)-Hölder continuous linearization near hyperbolic fixed points in \(\mathbb{R}^n\)
The paper is devoted to the study of problems related to regularity of a local linearization for a smooth nonlinear map \(T:\mathbb R^n\to\mathbb R^n\) near its hyperbolic fixed point. Recall that the classical Grobman-Hartman theorem guarantees only the existence of a continuous linearization \(H\) of the map \(T\) near the hyperbolic fixed point \(x=0\). On the other hand, the nonexistence results for \(C^1\)-linearization are also well known [see e.g. \textit{P. Hartman}, Ordinary differential equations, Wiley, New York (1964; Zbl 0125.32102). In the present paper the possibility of \(\sigma\)-Hölder continuous linearization of any \(C^{1,1}\)-map \(T\) near its hyperbolic fixed point \(x=0\), where the exponent \(\sigma\in(0,1)\) can be explicitly expressed in terms of the spectrum of \(D_xT(0)\), is proved. Using this explicit expression the author formulates conditions, which guarantee the existence of \(\sigma\)-Hölder continuous linearization with any \(\sigma\in(0,1)\). In particular, it is shown that for \(3\)-dimensional maps \(T:\mathbb R^3\to\mathbb R^3\) the \(\sigma\)-Hölder continuous linearization with any \(\sigma\in (0,1)\) exists without additional restrictions on \(T\).
scientific article; zbMATH DE number 1454734