### Abstract

Ge, Rusjan, and Zweifel introduced a binary tree which represents all the periodic windows in the chaotic regime of iterated one-dimensional unimodal maps. We consider the scaling behavior in a modified tree which takes into account the self-similarity of the window structure. A nonuniversal geometric convergence of the associated superstable parameter values towards a Misiurewicz point is observed for almost all binary sequences with periodic tails. For these sequences the window period grows arithmetically down the binary tree. There are an infinite number of exceptional sequences, however, for which the growth of the window period is faster. Numerical studies with a quadratic maximum suggest more rapid than geometric scaling of the superstable parameter values for such sequences.

Original language | English |
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Pages (from-to) | 643-668 |

Number of pages | 26 |

Journal | Journal of Statistical Physics |

Volume | 75 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 1 May 1994 |

MoE publication type | Not Eligible |

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### Keywords

- binary tree
- chaos
- Misiurewicz point
- periodic window
- scaling
- Unimodal map

### Cite this

*Journal of Statistical Physics*,

*75*(3-4), 643-668. https://doi.org/10.1007/BF02186875