Known Results and Open Problems on C1 linearization in Banach Spaces

Abstract

Abstract. The purpose of this paper is to review the results obtained by the authors on linearization of dynamical systems in infinite dimensional Banach spaces, especially in the C 1 case, and also to present some open problems that we believe that are still important for the understanding of the theory.The purpose of the present paper is to make a review of results obtained by the authors in the field of linearization of dynamical systems in infinite dimensional Banach spaces, with a focuss on C 1 linearization of contractions, with the intention of stating in the right context a number of open problems we believe it would be worth to be solved.1991 Mathematics Subject Classification. AMS Classification. Primary: 37C15, 35B05 Secondary: 34C20, 37D05.

Differentiability with respect to parameters in global smooth linearization

“…ElBialy [2] and by W. Zhang, W. Zhang and W. Jarczyk [10] on resonant fixed points and on sharp regularity estimates, respectively. We also address to our previous paper [9] for more references and some open problems. In our previous work [8] we studied the continuous dependence on parameters for C 0 linearization, and our present paper is strongly influenced by these previous results.…”

An example on Lyapunov stability and linearization

“…We want to say that our work started after attending the stimulating talk Remarks on We want also to say that we have been interested in linearization in infinite dimensions for a number of years, starting with our work [7]. See a summary of our work in [9]. And the present paper is not the first time we find an example of a phenomenon in this field that cannot happen in finite dimensions, as it was the example we studied in [8].…”