Renormalization tools to study the loss of stability in the Area Preserving Maps

Abstract

The renormalization method for area preserving maps was introduced by R. S. MacKay in 1982 to study critical invariant circles. Similar ideas appear in the study of critical phenomena in physics (MacKay 1983). The renormalization scenario for the breakdown of golden invarian circles in the twist map is described in this paper. In the first part we describe the renormalization methods introduced by MacKay. In the second part, the obstruction criterion (Olvera & Simó 1987) is described and we show that the renormalization group and the obstruction criterion can work together (De la Llave & Olvera 2006).