| 概要 | We study dynamics of the strongly dissipative Henon map at the first
bifurcation parameter where the uniform hyperbolicityis destroyed by the
formation of tangencies inside the limit set. We effect a multifractal
formalism, i.e. decompose the phase space into level sets of time
averages of a continuous function, and give partial descriptions of the
Birkhoff spectrum which encodes this decomposition. Using a canonical
induced map, we derive a formula for the Hausdorff dimension of the
level sets in terms of Lyapunov exponent and entropy of invariant
probability measures, and then show that the spectrum is continuous. The
main idea is to relate each level set to embedded hyperbolic sets.
References:
[1] Y. M. Chung and H. Takahasi: Multifractal formalism for
Benedicks-Carleson quadratic maps. Ergodic Theory and Dynamical Systems,
to appear
[2] S. Senti and H. Takahasi: Equilibrium measures for the Henon mapat
the first bifurcation. (arXiv:1209.2224)
[3] S. Senti and H. Takahasi: Equilibrium measures for the Henon map at
the first bifurcation: uniqueness and geometric/statistical properties.
(arXiv:1110.0601)
[4] H. Takahasi: Prevalent dynamics at the first bifurcation of
Henon-like families. Communications in Mathematical Physics (2012) 312,
37-85.
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