### Abstract

For a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we effect a multifractal analysis, i.e. decompose the set of non-wandering points on the unstable manifold into level sets of an unstable Lyapunov exponent, and give a partial description of the Lyapunov spectrum which encodes this decomposition. We derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures, and show the continuity of the Lyapunov spectrum. We also show that the set of points for which the unstable Lyapunov exponents do not exist carries the full Hausdorff dimension.

Original language | English |
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Pages (from-to) | 1-33 |

Number of pages | 33 |

Journal | Ergodic Theory and Dynamical Systems |

DOIs | |

Publication status | Accepted/In press - 2016 Nov 10 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Lyapunov spectrum for Hénon-like maps at the first bifurcation.** / Takahasi, Hiroki.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Lyapunov spectrum for Hénon-like maps at the first bifurcation

AU - Takahasi, Hiroki

PY - 2016/11/10

Y1 - 2016/11/10

N2 - For a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we effect a multifractal analysis, i.e. decompose the set of non-wandering points on the unstable manifold into level sets of an unstable Lyapunov exponent, and give a partial description of the Lyapunov spectrum which encodes this decomposition. We derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures, and show the continuity of the Lyapunov spectrum. We also show that the set of points for which the unstable Lyapunov exponents do not exist carries the full Hausdorff dimension.

AB - For a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we effect a multifractal analysis, i.e. decompose the set of non-wandering points on the unstable manifold into level sets of an unstable Lyapunov exponent, and give a partial description of the Lyapunov spectrum which encodes this decomposition. We derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures, and show the continuity of the Lyapunov spectrum. We also show that the set of points for which the unstable Lyapunov exponents do not exist carries the full Hausdorff dimension.

UR - http://www.scopus.com/inward/record.url?scp=84995474889&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84995474889&partnerID=8YFLogxK

U2 - 10.1017/etds.2016.65

DO - 10.1017/etds.2016.65

M3 - Article

AN - SCOPUS:84995474889

SP - 1

EP - 33

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

ER -