A Proof of the Persistence of Anti-integrable States for Three-Dimensional Quadratic Diffeomorphisms

Abstract

Three-dimensional quadratic diffeomorphisms with quadratic inverse generically have five independent parameters. When some parameters approach infinity, the diffeomorphisms may exhibit a so-called anti-integrable limit in the traditional sense of Aubry and Abramovici. That is, the dynamics of the diffeomorphisms reduce to symbolic dynamics on a finite number of symbols. However, the diffeomorphisms may reduce to quadratic correspondences when parameters approach infinity, and the traditional anti-integrable limit does not deal with this situation. Meiss asked what about an anti-integrable limit for it. Remarkable progress was achieved very recently by the work of Hampton and Meiss [SIAM J. Appl. Dyn. Syst. 21 (2022), pp. 650--675]. Using the contraction mapping theorem, they showed there is a bijection between the anti-integrable states and the sequences of branches of a quadratic correspondence. They also showed that an anti-integrable state can be continued to a genuine orbit of the three-dimensional diffeomorphism. This paper aims to contribute to the progress, by means of the implicit function theorem. We shall show that, under a slightly more restricted condition than that imposed by Hampton and Meiss, the bijection indeed is a topological conjugacy and establish the uniform hyperbolicity of the continued genuine orbits.