On a Question of J.D. Meiss Concerning Anti-integrability 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 the so-called anti-integrability in the traditional sense of Aubry and Abramovici. That is, the dynamics of the diffeomorphisms reduce to symbolic dynamics on 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. A remarkable progress was achieved very recently by Meiss himself and his student Hampton [SIAM J. Appl. Dyn. Syst. 21 (2022), pp. 650--675]. Under some conditions, 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 the progress, by means of the implicit function theorem. We shall show that the bijection indeed is a topological conjugacy and establish the uniform hyperbolicity of the continued genuine orbits.