Abstract
Since the Hénon 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 onR 2 whenb∈0, we can regard the periodic orbit ofH a, 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 Hénon family, we consider a question: what kind of relation exists, when we classify the periodic orbits of the Hénon 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.
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Dedicated to Professor Kenichi Shiraiwa on his 60th Birthday
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Sannami, A. A topological classification of the periodic orbits of the Hénon family. Japan J. Appl. Math. 6, 291–330 (1989). https://doi.org/10.1007/BF03167883
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DOI: https://doi.org/10.1007/BF03167883