## Abstract

Let K be a complete, algebraically closed, non-Archimedean valued field, and let P ^{1} denote the Berkovich projective line over K. The Lyapunov exponent for a rational map Φ ε K(z) of degree d ≥ 2 measures the exponential rate of growth along a typical orbit of Φ. When Φ is defined over C, the Lyapunov exponent is bounded below by 1/2 log d. In this article, we give a lower bound for L(Φ) for maps Φ defined over non-Archimedean fields K. The bound depends only on the degree d and the Lipschitz constant of Φ. For maps Φ whose Julia sets satisfy a certain boundedness condition, we are able to remove the dependence on the Lipschitz constant.

Original language | English (US) |
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Pages (from-to) | 6025-6046 |

Number of pages | 22 |

Journal | Transactions of the American Mathematical Society |

Volume | 371 |

Issue number | 9 |

DOIs | |

State | Published - Jan 1 2019 |

## Keywords

- Distortion
- Lower bound
- Lyapunov exponent
- Non-archimedean
- Rational map

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics