Year: 2019
Type: preprint
Abstract: A well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by P. Hartman in 1960 for operators on finite-dimensional spaces. The general case was independently obtained by J. Palis and C. Pugh around 1968. We will exhibit examples of structurally stable operators that are not hyperbolic, thereby showing that the converse of the above-mentioned result is false in general. We will also prove that an invertible operator on a Banach space is hyperbolic if and only if it is expansive and has the shadowing property. Moreover, we will show that if a structurally stable operator is expansive, then it must be uniformly expansive. Finally, we will characterize the weighted shifts on the spaces $c_0(\mathbb{Z})$ and $\ell_p(\mathbb{Z})$ ($1 \leq p < \infty$) that satisfy the shadowing property.
Source: arXiv (Cornell University)
Authors: Bernardes, Nilson C., Messaoudi, Ali
Institution: Universidade Federal do Rio de Janeiro
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G2927333014, G872243465
OpenAlex ID: W2913036314