A topological classification of the periodic orbits of the Hénon family

Abstract

Since the Henon map,\(H_{a,b} \left[ \begin{array}{l} x \\ y \\ \end{array} \right] = \left[ \begin{array}{l} y + 1 - ax^2 \\ bx \\ \end{array} \right]\), is a diffeomorphism onR2 whenb∈0, we can regard the periodic orbit ofHa, b as a “braid”. It is shown that two homeomorphisms on a disk are isotopic, preserving their periodic orbits, if and only if the corresponding braids are conjugate with each other (”r-conjugate”, when they are orientation reversing). Being motivated by the global bifurcation structure on the 2-parameter space of the Henon family, we consider a question: what kind of relation exists, when we classify the periodic orbits of the Henon family using such isotopy? By the above fact, we investigate the conjugacy relation between corresponding braids. A special relation pattern in a certain class of periodic orbits is obtained. This pattern has a self-similar like structure related to a hyperbolic set of period 2.